I am done

rl/Lib/site-packages/torch/distributions/__init__.py

@@ -0,0 +1,172 @@
+r"""
+The ``distributions`` package contains parameterizable probability distributions
+and sampling functions. This allows the construction of stochastic computation
+graphs and stochastic gradient estimators for optimization. This package
+generally follows the design of the `TensorFlow Distributions`_ package.
+
+.. _`TensorFlow Distributions`:
+    https://arxiv.org/abs/1711.10604
+
+It is not possible to directly backpropagate through random samples. However,
+there are two main methods for creating surrogate functions that can be
+backpropagated through. These are the score function estimator/likelihood ratio
+estimator/REINFORCE and the pathwise derivative estimator. REINFORCE is commonly
+seen as the basis for policy gradient methods in reinforcement learning, and the
+pathwise derivative estimator is commonly seen in the reparameterization trick
+in variational autoencoders. Whilst the score function only requires the value
+of samples :math:`f(x)`, the pathwise derivative requires the derivative
+:math:`f'(x)`. The next sections discuss these two in a reinforcement learning
+example. For more details see
+`Gradient Estimation Using Stochastic Computation Graphs`_ .
+
+.. _`Gradient Estimation Using Stochastic Computation Graphs`:
+     https://arxiv.org/abs/1506.05254
+
+Score function
+^^^^^^^^^^^^^^
+
+When the probability density function is differentiable with respect to its
+parameters, we only need :meth:`~torch.distributions.Distribution.sample` and
+:meth:`~torch.distributions.Distribution.log_prob` to implement REINFORCE:
+
+.. math::
+
+    \Delta\theta  = \alpha r \frac{\partial\log p(a|\pi^\theta(s))}{\partial\theta}
+
+where :math:`\theta` are the parameters, :math:`\alpha` is the learning rate,
+:math:`r` is the reward and :math:`p(a|\pi^\theta(s))` is the probability of
+taking action :math:`a` in state :math:`s` given policy :math:`\pi^\theta`.
+
+In practice we would sample an action from the output of a network, apply this
+action in an environment, and then use ``log_prob`` to construct an equivalent
+loss function. Note that we use a negative because optimizers use gradient
+descent, whilst the rule above assumes gradient ascent. With a categorical
+policy, the code for implementing REINFORCE would be as follows::
+
+    probs = policy_network(state)
+    # Note that this is equivalent to what used to be called multinomial
+    m = Categorical(probs)
+    action = m.sample()
+    next_state, reward = env.step(action)
+    loss = -m.log_prob(action) * reward
+    loss.backward()
+
+Pathwise derivative
+^^^^^^^^^^^^^^^^^^^
+
+The other way to implement these stochastic/policy gradients would be to use the
+reparameterization trick from the
+:meth:`~torch.distributions.Distribution.rsample` method, where the
+parameterized random variable can be constructed via a parameterized
+deterministic function of a parameter-free random variable. The reparameterized
+sample therefore becomes differentiable. The code for implementing the pathwise
+derivative would be as follows::
+
+    params = policy_network(state)
+    m = Normal(*params)
+    # Any distribution with .has_rsample == True could work based on the application
+    action = m.rsample()
+    next_state, reward = env.step(action)  # Assuming that reward is differentiable
+    loss = -reward
+    loss.backward()
+"""
+
+from . import transforms
+from .bernoulli import Bernoulli
+from .beta import Beta
+from .binomial import Binomial
+from .categorical import Categorical
+from .cauchy import Cauchy
+from .chi2 import Chi2
+from .constraint_registry import biject_to, transform_to
+from .continuous_bernoulli import ContinuousBernoulli
+from .dirichlet import Dirichlet
+from .distribution import Distribution
+from .exp_family import ExponentialFamily
+from .exponential import Exponential
+from .fishersnedecor import FisherSnedecor
+from .gamma import Gamma
+from .geometric import Geometric
+from .gumbel import Gumbel
+from .half_cauchy import HalfCauchy
+from .half_normal import HalfNormal
+from .independent import Independent
+from .inverse_gamma import InverseGamma
+from .kl import _add_kl_info, kl_divergence, register_kl
+from .kumaraswamy import Kumaraswamy
+from .laplace import Laplace
+from .lkj_cholesky import LKJCholesky
+from .log_normal import LogNormal
+from .logistic_normal import LogisticNormal
+from .lowrank_multivariate_normal import LowRankMultivariateNormal
+from .mixture_same_family import MixtureSameFamily
+from .multinomial import Multinomial
+from .multivariate_normal import MultivariateNormal
+from .negative_binomial import NegativeBinomial
+from .normal import Normal
+from .one_hot_categorical import OneHotCategorical, OneHotCategoricalStraightThrough
+from .pareto import Pareto
+from .poisson import Poisson
+from .relaxed_bernoulli import RelaxedBernoulli
+from .relaxed_categorical import RelaxedOneHotCategorical
+from .studentT import StudentT
+from .transformed_distribution import TransformedDistribution
+from .transforms import *  # noqa: F403
+from .uniform import Uniform
+from .von_mises import VonMises
+from .weibull import Weibull
+from .wishart import Wishart
+
+
+_add_kl_info()
+del _add_kl_info
+
+__all__ = [
+    "Bernoulli",
+    "Beta",
+    "Binomial",
+    "Categorical",
+    "Cauchy",
+    "Chi2",
+    "ContinuousBernoulli",
+    "Dirichlet",
+    "Distribution",
+    "Exponential",
+    "ExponentialFamily",
+    "FisherSnedecor",
+    "Gamma",
+    "Geometric",
+    "Gumbel",
+    "HalfCauchy",
+    "HalfNormal",
+    "Independent",
+    "InverseGamma",
+    "Kumaraswamy",
+    "LKJCholesky",
+    "Laplace",
+    "LogNormal",
+    "LogisticNormal",
+    "LowRankMultivariateNormal",
+    "MixtureSameFamily",
+    "Multinomial",
+    "MultivariateNormal",
+    "NegativeBinomial",
+    "Normal",
+    "OneHotCategorical",
+    "OneHotCategoricalStraightThrough",
+    "Pareto",
+    "RelaxedBernoulli",
+    "RelaxedOneHotCategorical",
+    "StudentT",
+    "Poisson",
+    "Uniform",
+    "VonMises",
+    "Weibull",
+    "Wishart",
+    "TransformedDistribution",
+    "biject_to",
+    "kl_divergence",
+    "register_kl",
+    "transform_to",
+]
+__all__.extend(transforms.__all__)

rl/Lib/site-packages/torch/distributions/bernoulli.py

@@ -0,0 +1,132 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch import nan
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import (
+    broadcast_all,
+    lazy_property,
+    logits_to_probs,
+    probs_to_logits,
+)
+from torch.nn.functional import binary_cross_entropy_with_logits
+
+
+__all__ = ["Bernoulli"]
+
+
+class Bernoulli(ExponentialFamily):
+    r"""
+    Creates a Bernoulli distribution parameterized by :attr:`probs`
+    or :attr:`logits` (but not both).
+
+    Samples are binary (0 or 1). They take the value `1` with probability `p`
+    and `0` with probability `1 - p`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Bernoulli(torch.tensor([0.3]))
+        >>> m.sample()  # 30% chance 1; 70% chance 0
+        tensor([ 0.])
+
+    Args:
+        probs (Number, Tensor): the probability of sampling `1`
+        logits (Number, Tensor): the log-odds of sampling `1`
+    """
+    arg_constraints = {"probs": constraints.unit_interval, "logits": constraints.real}
+    support = constraints.boolean
+    has_enumerate_support = True
+    _mean_carrier_measure = 0
+
+    def __init__(self, probs=None, logits=None, validate_args=None):
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            is_scalar = isinstance(probs, Number)
+            (self.probs,) = broadcast_all(probs)
+        else:
+            is_scalar = isinstance(logits, Number)
+            (self.logits,) = broadcast_all(logits)
+        self._param = self.probs if probs is not None else self.logits
+        if is_scalar:
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self._param.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Bernoulli, _instance)
+        batch_shape = torch.Size(batch_shape)
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(batch_shape)
+            new._param = new.probs
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(batch_shape)
+            new._param = new.logits
+        super(Bernoulli, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._param.new(*args, **kwargs)
+
+    @property
+    def mean(self):
+        return self.probs
+
+    @property
+    def mode(self):
+        mode = (self.probs >= 0.5).to(self.probs)
+        mode[self.probs == 0.5] = nan
+        return mode
+
+    @property
+    def variance(self):
+        return self.probs * (1 - self.probs)
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs, is_binary=True)
+
+    @lazy_property
+    def probs(self):
+        return logits_to_probs(self.logits, is_binary=True)
+
+    @property
+    def param_shape(self):
+        return self._param.size()
+
+    def sample(self, sample_shape=torch.Size()):
+        shape = self._extended_shape(sample_shape)
+        with torch.no_grad():
+            return torch.bernoulli(self.probs.expand(shape))
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        logits, value = broadcast_all(self.logits, value)
+        return -binary_cross_entropy_with_logits(logits, value, reduction="none")
+
+    def entropy(self):
+        return binary_cross_entropy_with_logits(
+            self.logits, self.probs, reduction="none"
+        )
+
+    def enumerate_support(self, expand=True):
+        values = torch.arange(2, dtype=self._param.dtype, device=self._param.device)
+        values = values.view((-1,) + (1,) * len(self._batch_shape))
+        if expand:
+            values = values.expand((-1,) + self._batch_shape)
+        return values
+
+    @property
+    def _natural_params(self):
+        return (torch.logit(self.probs),)
+
+    def _log_normalizer(self, x):
+        return torch.log1p(torch.exp(x))

rl/Lib/site-packages/torch/distributions/beta.py

@@ -0,0 +1,110 @@
+# mypy: allow-untyped-defs
+from numbers import Number, Real
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.dirichlet import Dirichlet
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Beta"]
+
+
+class Beta(ExponentialFamily):
+    r"""
+    Beta distribution parameterized by :attr:`concentration1` and :attr:`concentration0`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Beta(torch.tensor([0.5]), torch.tensor([0.5]))
+        >>> m.sample()  # Beta distributed with concentration concentration1 and concentration0
+        tensor([ 0.1046])
+
+    Args:
+        concentration1 (float or Tensor): 1st concentration parameter of the distribution
+            (often referred to as alpha)
+        concentration0 (float or Tensor): 2nd concentration parameter of the distribution
+            (often referred to as beta)
+    """
+    arg_constraints = {
+        "concentration1": constraints.positive,
+        "concentration0": constraints.positive,
+    }
+    support = constraints.unit_interval
+    has_rsample = True
+
+    def __init__(self, concentration1, concentration0, validate_args=None):
+        if isinstance(concentration1, Real) and isinstance(concentration0, Real):
+            concentration1_concentration0 = torch.tensor(
+                [float(concentration1), float(concentration0)]
+            )
+        else:
+            concentration1, concentration0 = broadcast_all(
+                concentration1, concentration0
+            )
+            concentration1_concentration0 = torch.stack(
+                [concentration1, concentration0], -1
+            )
+        self._dirichlet = Dirichlet(
+            concentration1_concentration0, validate_args=validate_args
+        )
+        super().__init__(self._dirichlet._batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Beta, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new._dirichlet = self._dirichlet.expand(batch_shape)
+        super(Beta, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def mean(self):
+        return self.concentration1 / (self.concentration1 + self.concentration0)
+
+    @property
+    def mode(self):
+        return self._dirichlet.mode[..., 0]
+
+    @property
+    def variance(self):
+        total = self.concentration1 + self.concentration0
+        return self.concentration1 * self.concentration0 / (total.pow(2) * (total + 1))
+
+    def rsample(self, sample_shape: _size = ()) -> torch.Tensor:
+        return self._dirichlet.rsample(sample_shape).select(-1, 0)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        heads_tails = torch.stack([value, 1.0 - value], -1)
+        return self._dirichlet.log_prob(heads_tails)
+
+    def entropy(self):
+        return self._dirichlet.entropy()
+
+    @property
+    def concentration1(self):
+        result = self._dirichlet.concentration[..., 0]
+        if isinstance(result, Number):
+            return torch.tensor([result])
+        else:
+            return result
+
+    @property
+    def concentration0(self):
+        result = self._dirichlet.concentration[..., 1]
+        if isinstance(result, Number):
+            return torch.tensor([result])
+        else:
+            return result
+
+    @property
+    def _natural_params(self):
+        return (self.concentration1, self.concentration0)
+
+    def _log_normalizer(self, x, y):
+        return torch.lgamma(x) + torch.lgamma(y) - torch.lgamma(x + y)

rl/Lib/site-packages/torch/distributions/binomial.py

@@ -0,0 +1,167 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import (
+    broadcast_all,
+    lazy_property,
+    logits_to_probs,
+    probs_to_logits,
+)
+
+
+__all__ = ["Binomial"]
+
+
+def _clamp_by_zero(x):
+    # works like clamp(x, min=0) but has grad at 0 is 0.5
+    return (x.clamp(min=0) + x - x.clamp(max=0)) / 2
+
+
+class Binomial(Distribution):
+    r"""
+    Creates a Binomial distribution parameterized by :attr:`total_count` and
+    either :attr:`probs` or :attr:`logits` (but not both). :attr:`total_count` must be
+    broadcastable with :attr:`probs`/:attr:`logits`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Binomial(100, torch.tensor([0 , .2, .8, 1]))
+        >>> x = m.sample()
+        tensor([   0.,   22.,   71.,  100.])
+
+        >>> m = Binomial(torch.tensor([[5.], [10.]]), torch.tensor([0.5, 0.8]))
+        >>> x = m.sample()
+        tensor([[ 4.,  5.],
+                [ 7.,  6.]])
+
+    Args:
+        total_count (int or Tensor): number of Bernoulli trials
+        probs (Tensor): Event probabilities
+        logits (Tensor): Event log-odds
+    """
+    arg_constraints = {
+        "total_count": constraints.nonnegative_integer,
+        "probs": constraints.unit_interval,
+        "logits": constraints.real,
+    }
+    has_enumerate_support = True
+
+    def __init__(self, total_count=1, probs=None, logits=None, validate_args=None):
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            (
+                self.total_count,
+                self.probs,
+            ) = broadcast_all(total_count, probs)
+            self.total_count = self.total_count.type_as(self.probs)
+        else:
+            (
+                self.total_count,
+                self.logits,
+            ) = broadcast_all(total_count, logits)
+            self.total_count = self.total_count.type_as(self.logits)
+
+        self._param = self.probs if probs is not None else self.logits
+        batch_shape = self._param.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Binomial, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.total_count = self.total_count.expand(batch_shape)
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(batch_shape)
+            new._param = new.probs
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(batch_shape)
+            new._param = new.logits
+        super(Binomial, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._param.new(*args, **kwargs)
+
+    @constraints.dependent_property(is_discrete=True, event_dim=0)
+    def support(self):
+        return constraints.integer_interval(0, self.total_count)
+
+    @property
+    def mean(self):
+        return self.total_count * self.probs
+
+    @property
+    def mode(self):
+        return ((self.total_count + 1) * self.probs).floor().clamp(max=self.total_count)
+
+    @property
+    def variance(self):
+        return self.total_count * self.probs * (1 - self.probs)
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs, is_binary=True)
+
+    @lazy_property
+    def probs(self):
+        return logits_to_probs(self.logits, is_binary=True)
+
+    @property
+    def param_shape(self):
+        return self._param.size()
+
+    def sample(self, sample_shape=torch.Size()):
+        shape = self._extended_shape(sample_shape)
+        with torch.no_grad():
+            return torch.binomial(
+                self.total_count.expand(shape), self.probs.expand(shape)
+            )
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        log_factorial_n = torch.lgamma(self.total_count + 1)
+        log_factorial_k = torch.lgamma(value + 1)
+        log_factorial_nmk = torch.lgamma(self.total_count - value + 1)
+        # k * log(p) + (n - k) * log(1 - p) = k * (log(p) - log(1 - p)) + n * log(1 - p)
+        #     (case logit < 0)              = k * logit - n * log1p(e^logit)
+        #     (case logit > 0)              = k * logit - n * (log(p) - log(1 - p)) + n * log(p)
+        #                                   = k * logit - n * logit - n * log1p(e^-logit)
+        #     (merge two cases)             = k * logit - n * max(logit, 0) - n * log1p(e^-|logit|)
+        normalize_term = (
+            self.total_count * _clamp_by_zero(self.logits)
+            + self.total_count * torch.log1p(torch.exp(-torch.abs(self.logits)))
+            - log_factorial_n
+        )
+        return (
+            value * self.logits - log_factorial_k - log_factorial_nmk - normalize_term
+        )
+
+    def entropy(self):
+        total_count = int(self.total_count.max())
+        if not self.total_count.min() == total_count:
+            raise NotImplementedError(
+                "Inhomogeneous total count not supported by `entropy`."
+            )
+
+        log_prob = self.log_prob(self.enumerate_support(False))
+        return -(torch.exp(log_prob) * log_prob).sum(0)
+
+    def enumerate_support(self, expand=True):
+        total_count = int(self.total_count.max())
+        if not self.total_count.min() == total_count:
+            raise NotImplementedError(
+                "Inhomogeneous total count not supported by `enumerate_support`."
+            )
+        values = torch.arange(
+            1 + total_count, dtype=self._param.dtype, device=self._param.device
+        )
+        values = values.view((-1,) + (1,) * len(self._batch_shape))
+        if expand:
+            values = values.expand((-1,) + self._batch_shape)
+        return values

rl/Lib/site-packages/torch/distributions/categorical.py

@@ -0,0 +1,157 @@
+# mypy: allow-untyped-defs
+import torch
+from torch import nan
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import lazy_property, logits_to_probs, probs_to_logits
+
+
+__all__ = ["Categorical"]
+
+
+class Categorical(Distribution):
+    r"""
+    Creates a categorical distribution parameterized by either :attr:`probs` or
+    :attr:`logits` (but not both).
+
+    .. note::
+        It is equivalent to the distribution that :func:`torch.multinomial`
+        samples from.
+
+    Samples are integers from :math:`\{0, \ldots, K-1\}` where `K` is ``probs.size(-1)``.
+
+    If `probs` is 1-dimensional with length-`K`, each element is the relative probability
+    of sampling the class at that index.
+
+    If `probs` is N-dimensional, the first N-1 dimensions are treated as a batch of
+    relative probability vectors.
+
+    .. note:: The `probs` argument must be non-negative, finite and have a non-zero sum,
+              and it will be normalized to sum to 1 along the last dimension. :attr:`probs`
+              will return this normalized value.
+              The `logits` argument will be interpreted as unnormalized log probabilities
+              and can therefore be any real number. It will likewise be normalized so that
+              the resulting probabilities sum to 1 along the last dimension. :attr:`logits`
+              will return this normalized value.
+
+    See also: :func:`torch.multinomial`
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Categorical(torch.tensor([ 0.25, 0.25, 0.25, 0.25 ]))
+        >>> m.sample()  # equal probability of 0, 1, 2, 3
+        tensor(3)
+
+    Args:
+        probs (Tensor): event probabilities
+        logits (Tensor): event log probabilities (unnormalized)
+    """
+    arg_constraints = {"probs": constraints.simplex, "logits": constraints.real_vector}
+    has_enumerate_support = True
+
+    def __init__(self, probs=None, logits=None, validate_args=None):
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            if probs.dim() < 1:
+                raise ValueError("`probs` parameter must be at least one-dimensional.")
+            self.probs = probs / probs.sum(-1, keepdim=True)
+        else:
+            if logits.dim() < 1:
+                raise ValueError("`logits` parameter must be at least one-dimensional.")
+            # Normalize
+            self.logits = logits - logits.logsumexp(dim=-1, keepdim=True)
+        self._param = self.probs if probs is not None else self.logits
+        self._num_events = self._param.size()[-1]
+        batch_shape = (
+            self._param.size()[:-1] if self._param.ndimension() > 1 else torch.Size()
+        )
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Categorical, _instance)
+        batch_shape = torch.Size(batch_shape)
+        param_shape = batch_shape + torch.Size((self._num_events,))
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(param_shape)
+            new._param = new.probs
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(param_shape)
+            new._param = new.logits
+        new._num_events = self._num_events
+        super(Categorical, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._param.new(*args, **kwargs)
+
+    @constraints.dependent_property(is_discrete=True, event_dim=0)
+    def support(self):
+        return constraints.integer_interval(0, self._num_events - 1)
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs)
+
+    @lazy_property
+    def probs(self):
+        return logits_to_probs(self.logits)
+
+    @property
+    def param_shape(self):
+        return self._param.size()
+
+    @property
+    def mean(self):
+        return torch.full(
+            self._extended_shape(),
+            nan,
+            dtype=self.probs.dtype,
+            device=self.probs.device,
+        )
+
+    @property
+    def mode(self):
+        return self.probs.argmax(axis=-1)
+
+    @property
+    def variance(self):
+        return torch.full(
+            self._extended_shape(),
+            nan,
+            dtype=self.probs.dtype,
+            device=self.probs.device,
+        )
+
+    def sample(self, sample_shape=torch.Size()):
+        if not isinstance(sample_shape, torch.Size):
+            sample_shape = torch.Size(sample_shape)
+        probs_2d = self.probs.reshape(-1, self._num_events)
+        samples_2d = torch.multinomial(probs_2d, sample_shape.numel(), True).T
+        return samples_2d.reshape(self._extended_shape(sample_shape))
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        value = value.long().unsqueeze(-1)
+        value, log_pmf = torch.broadcast_tensors(value, self.logits)
+        value = value[..., :1]
+        return log_pmf.gather(-1, value).squeeze(-1)
+
+    def entropy(self):
+        min_real = torch.finfo(self.logits.dtype).min
+        logits = torch.clamp(self.logits, min=min_real)
+        p_log_p = logits * self.probs
+        return -p_log_p.sum(-1)
+
+    def enumerate_support(self, expand=True):
+        num_events = self._num_events
+        values = torch.arange(num_events, dtype=torch.long, device=self._param.device)
+        values = values.view((-1,) + (1,) * len(self._batch_shape))
+        if expand:
+            values = values.expand((-1,) + self._batch_shape)
+        return values

rl/Lib/site-packages/torch/distributions/cauchy.py

@@ -0,0 +1,93 @@
+# mypy: allow-untyped-defs
+import math
+from numbers import Number
+
+import torch
+from torch import inf, nan
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Cauchy"]
+
+
+class Cauchy(Distribution):
+    r"""
+    Samples from a Cauchy (Lorentz) distribution. The distribution of the ratio of
+    independent normally distributed random variables with means `0` follows a
+    Cauchy distribution.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Cauchy(torch.tensor([0.0]), torch.tensor([1.0]))
+        >>> m.sample()  # sample from a Cauchy distribution with loc=0 and scale=1
+        tensor([ 2.3214])
+
+    Args:
+        loc (float or Tensor): mode or median of the distribution.
+        scale (float or Tensor): half width at half maximum.
+    """
+    arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
+    support = constraints.real
+    has_rsample = True
+
+    def __init__(self, loc, scale, validate_args=None):
+        self.loc, self.scale = broadcast_all(loc, scale)
+        if isinstance(loc, Number) and isinstance(scale, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.loc.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Cauchy, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.loc = self.loc.expand(batch_shape)
+        new.scale = self.scale.expand(batch_shape)
+        super(Cauchy, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def mean(self):
+        return torch.full(
+            self._extended_shape(), nan, dtype=self.loc.dtype, device=self.loc.device
+        )
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @property
+    def variance(self):
+        return torch.full(
+            self._extended_shape(), inf, dtype=self.loc.dtype, device=self.loc.device
+        )
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        eps = self.loc.new(shape).cauchy_()
+        return self.loc + eps * self.scale
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return (
+            -math.log(math.pi)
+            - self.scale.log()
+            - (((value - self.loc) / self.scale) ** 2).log1p()
+        )
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return torch.atan((value - self.loc) / self.scale) / math.pi + 0.5
+
+    def icdf(self, value):
+        return torch.tan(math.pi * (value - 0.5)) * self.scale + self.loc
+
+    def entropy(self):
+        return math.log(4 * math.pi) + self.scale.log()

rl/Lib/site-packages/torch/distributions/chi2.py

@@ -0,0 +1,35 @@
+# mypy: allow-untyped-defs
+from torch.distributions import constraints
+from torch.distributions.gamma import Gamma
+
+
+__all__ = ["Chi2"]
+
+
+class Chi2(Gamma):
+    r"""
+    Creates a Chi-squared distribution parameterized by shape parameter :attr:`df`.
+    This is exactly equivalent to ``Gamma(alpha=0.5*df, beta=0.5)``
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Chi2(torch.tensor([1.0]))
+        >>> m.sample()  # Chi2 distributed with shape df=1
+        tensor([ 0.1046])
+
+    Args:
+        df (float or Tensor): shape parameter of the distribution
+    """
+    arg_constraints = {"df": constraints.positive}
+
+    def __init__(self, df, validate_args=None):
+        super().__init__(0.5 * df, 0.5, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Chi2, _instance)
+        return super().expand(batch_shape, new)
+
+    @property
+    def df(self):
+        return self.concentration * 2

rl/Lib/site-packages/torch/distributions/constraint_registry.py

@@ -0,0 +1,294 @@
+# mypy: allow-untyped-defs
+r"""
+PyTorch provides two global :class:`ConstraintRegistry` objects that link
+:class:`~torch.distributions.constraints.Constraint` objects to
+:class:`~torch.distributions.transforms.Transform` objects. These objects both
+input constraints and return transforms, but they have different guarantees on
+bijectivity.
+
+1. ``biject_to(constraint)`` looks up a bijective
+   :class:`~torch.distributions.transforms.Transform` from ``constraints.real``
+   to the given ``constraint``. The returned transform is guaranteed to have
+   ``.bijective = True`` and should implement ``.log_abs_det_jacobian()``.
+2. ``transform_to(constraint)`` looks up a not-necessarily bijective
+   :class:`~torch.distributions.transforms.Transform` from ``constraints.real``
+   to the given ``constraint``. The returned transform is not guaranteed to
+   implement ``.log_abs_det_jacobian()``.
+
+The ``transform_to()`` registry is useful for performing unconstrained
+optimization on constrained parameters of probability distributions, which are
+indicated by each distribution's ``.arg_constraints`` dict. These transforms often
+overparameterize a space in order to avoid rotation; they are thus more
+suitable for coordinate-wise optimization algorithms like Adam::
+
+    loc = torch.zeros(100, requires_grad=True)
+    unconstrained = torch.zeros(100, requires_grad=True)
+    scale = transform_to(Normal.arg_constraints['scale'])(unconstrained)
+    loss = -Normal(loc, scale).log_prob(data).sum()
+
+The ``biject_to()`` registry is useful for Hamiltonian Monte Carlo, where
+samples from a probability distribution with constrained ``.support`` are
+propagated in an unconstrained space, and algorithms are typically rotation
+invariant.::
+
+    dist = Exponential(rate)
+    unconstrained = torch.zeros(100, requires_grad=True)
+    sample = biject_to(dist.support)(unconstrained)
+    potential_energy = -dist.log_prob(sample).sum()
+
+.. note::
+
+    An example where ``transform_to`` and ``biject_to`` differ is
+    ``constraints.simplex``: ``transform_to(constraints.simplex)`` returns a
+    :class:`~torch.distributions.transforms.SoftmaxTransform` that simply
+    exponentiates and normalizes its inputs; this is a cheap and mostly
+    coordinate-wise operation appropriate for algorithms like SVI. In
+    contrast, ``biject_to(constraints.simplex)`` returns a
+    :class:`~torch.distributions.transforms.StickBreakingTransform` that
+    bijects its input down to a one-fewer-dimensional space; this a more
+    expensive less numerically stable transform but is needed for algorithms
+    like HMC.
+
+The ``biject_to`` and ``transform_to`` objects can be extended by user-defined
+constraints and transforms using their ``.register()`` method either as a
+function on singleton constraints::
+
+    transform_to.register(my_constraint, my_transform)
+
+or as a decorator on parameterized constraints::
+
+    @transform_to.register(MyConstraintClass)
+    def my_factory(constraint):
+        assert isinstance(constraint, MyConstraintClass)
+        return MyTransform(constraint.param1, constraint.param2)
+
+You can create your own registry by creating a new :class:`ConstraintRegistry`
+object.
+"""
+
+import numbers
+
+from torch.distributions import constraints, transforms
+
+
+__all__ = [
+    "ConstraintRegistry",
+    "biject_to",
+    "transform_to",
+]
+
+
+class ConstraintRegistry:
+    """
+    Registry to link constraints to transforms.
+    """
+
+    def __init__(self):
+        self._registry = {}
+        super().__init__()
+
+    def register(self, constraint, factory=None):
+        """
+        Registers a :class:`~torch.distributions.constraints.Constraint`
+        subclass in this registry. Usage::
+
+            @my_registry.register(MyConstraintClass)
+            def construct_transform(constraint):
+                assert isinstance(constraint, MyConstraint)
+                return MyTransform(constraint.arg_constraints)
+
+        Args:
+            constraint (subclass of :class:`~torch.distributions.constraints.Constraint`):
+                A subclass of :class:`~torch.distributions.constraints.Constraint`, or
+                a singleton object of the desired class.
+            factory (Callable): A callable that inputs a constraint object and returns
+                a  :class:`~torch.distributions.transforms.Transform` object.
+        """
+        # Support use as decorator.
+        if factory is None:
+            return lambda factory: self.register(constraint, factory)
+
+        # Support calling on singleton instances.
+        if isinstance(constraint, constraints.Constraint):
+            constraint = type(constraint)
+
+        if not isinstance(constraint, type) or not issubclass(
+            constraint, constraints.Constraint
+        ):
+            raise TypeError(
+                f"Expected constraint to be either a Constraint subclass or instance, but got {constraint}"
+            )
+
+        self._registry[constraint] = factory
+        return factory
+
+    def __call__(self, constraint):
+        """
+        Looks up a transform to constrained space, given a constraint object.
+        Usage::
+
+            constraint = Normal.arg_constraints['scale']
+            scale = transform_to(constraint)(torch.zeros(1))  # constrained
+            u = transform_to(constraint).inv(scale)           # unconstrained
+
+        Args:
+            constraint (:class:`~torch.distributions.constraints.Constraint`):
+                A constraint object.
+
+        Returns:
+            A :class:`~torch.distributions.transforms.Transform` object.
+
+        Raises:
+            `NotImplementedError` if no transform has been registered.
+        """
+        # Look up by Constraint subclass.
+        try:
+            factory = self._registry[type(constraint)]
+        except KeyError:
+            raise NotImplementedError(
+                f"Cannot transform {type(constraint).__name__} constraints"
+            ) from None
+        return factory(constraint)
+
+
+biject_to = ConstraintRegistry()
+transform_to = ConstraintRegistry()
+
+
+################################################################################
+# Registration Table
+################################################################################
+
+
+@biject_to.register(constraints.real)
+@transform_to.register(constraints.real)
+def _transform_to_real(constraint):
+    return transforms.identity_transform
+
+
+@biject_to.register(constraints.independent)
+def _biject_to_independent(constraint):
+    base_transform = biject_to(constraint.base_constraint)
+    return transforms.IndependentTransform(
+        base_transform, constraint.reinterpreted_batch_ndims
+    )
+
+
+@transform_to.register(constraints.independent)
+def _transform_to_independent(constraint):
+    base_transform = transform_to(constraint.base_constraint)
+    return transforms.IndependentTransform(
+        base_transform, constraint.reinterpreted_batch_ndims
+    )
+
+
+@biject_to.register(constraints.positive)
+@biject_to.register(constraints.nonnegative)
+@transform_to.register(constraints.positive)
+@transform_to.register(constraints.nonnegative)
+def _transform_to_positive(constraint):
+    return transforms.ExpTransform()
+
+
+@biject_to.register(constraints.greater_than)
+@biject_to.register(constraints.greater_than_eq)
+@transform_to.register(constraints.greater_than)
+@transform_to.register(constraints.greater_than_eq)
+def _transform_to_greater_than(constraint):
+    return transforms.ComposeTransform(
+        [
+            transforms.ExpTransform(),
+            transforms.AffineTransform(constraint.lower_bound, 1),
+        ]
+    )
+
+
+@biject_to.register(constraints.less_than)
+@transform_to.register(constraints.less_than)
+def _transform_to_less_than(constraint):
+    return transforms.ComposeTransform(
+        [
+            transforms.ExpTransform(),
+            transforms.AffineTransform(constraint.upper_bound, -1),
+        ]
+    )
+
+
+@biject_to.register(constraints.interval)
+@biject_to.register(constraints.half_open_interval)
+@transform_to.register(constraints.interval)
+@transform_to.register(constraints.half_open_interval)
+def _transform_to_interval(constraint):
+    # Handle the special case of the unit interval.
+    lower_is_0 = (
+        isinstance(constraint.lower_bound, numbers.Number)
+        and constraint.lower_bound == 0
+    )
+    upper_is_1 = (
+        isinstance(constraint.upper_bound, numbers.Number)
+        and constraint.upper_bound == 1
+    )
+    if lower_is_0 and upper_is_1:
+        return transforms.SigmoidTransform()
+
+    loc = constraint.lower_bound
+    scale = constraint.upper_bound - constraint.lower_bound
+    return transforms.ComposeTransform(
+        [transforms.SigmoidTransform(), transforms.AffineTransform(loc, scale)]
+    )
+
+
+@biject_to.register(constraints.simplex)
+def _biject_to_simplex(constraint):
+    return transforms.StickBreakingTransform()
+
+
+@transform_to.register(constraints.simplex)
+def _transform_to_simplex(constraint):
+    return transforms.SoftmaxTransform()
+
+
+# TODO define a bijection for LowerCholeskyTransform
+@transform_to.register(constraints.lower_cholesky)
+def _transform_to_lower_cholesky(constraint):
+    return transforms.LowerCholeskyTransform()
+
+
+@transform_to.register(constraints.positive_definite)
+@transform_to.register(constraints.positive_semidefinite)
+def _transform_to_positive_definite(constraint):
+    return transforms.PositiveDefiniteTransform()
+
+
+@biject_to.register(constraints.corr_cholesky)
+@transform_to.register(constraints.corr_cholesky)
+def _transform_to_corr_cholesky(constraint):
+    return transforms.CorrCholeskyTransform()
+
+
+@biject_to.register(constraints.cat)
+def _biject_to_cat(constraint):
+    return transforms.CatTransform(
+        [biject_to(c) for c in constraint.cseq], constraint.dim, constraint.lengths
+    )
+
+
+@transform_to.register(constraints.cat)
+def _transform_to_cat(constraint):
+    return transforms.CatTransform(
+        [transform_to(c) for c in constraint.cseq], constraint.dim, constraint.lengths
+    )
+
+
+@biject_to.register(constraints.stack)
+def _biject_to_stack(constraint):
+    return transforms.StackTransform(
+        [biject_to(c) for c in constraint.cseq], constraint.dim
+    )
+
+
+@transform_to.register(constraints.stack)
+def _transform_to_stack(constraint):
+    return transforms.StackTransform(
+        [transform_to(c) for c in constraint.cseq], constraint.dim
+    )

rl/Lib/site-packages/torch/distributions/constraints.py

@@ -0,0 +1,681 @@
+# mypy: allow-untyped-defs
+r"""
+The following constraints are implemented:
+
+- ``constraints.boolean``
+- ``constraints.cat``
+- ``constraints.corr_cholesky``
+- ``constraints.dependent``
+- ``constraints.greater_than(lower_bound)``
+- ``constraints.greater_than_eq(lower_bound)``
+- ``constraints.independent(constraint, reinterpreted_batch_ndims)``
+- ``constraints.integer_interval(lower_bound, upper_bound)``
+- ``constraints.interval(lower_bound, upper_bound)``
+- ``constraints.less_than(upper_bound)``
+- ``constraints.lower_cholesky``
+- ``constraints.lower_triangular``
+- ``constraints.multinomial``
+- ``constraints.nonnegative``
+- ``constraints.nonnegative_integer``
+- ``constraints.one_hot``
+- ``constraints.positive_integer``
+- ``constraints.positive``
+- ``constraints.positive_semidefinite``
+- ``constraints.positive_definite``
+- ``constraints.real_vector``
+- ``constraints.real``
+- ``constraints.simplex``
+- ``constraints.symmetric``
+- ``constraints.stack``
+- ``constraints.square``
+- ``constraints.symmetric``
+- ``constraints.unit_interval``
+"""
+
+import torch
+
+
+__all__ = [
+    "Constraint",
+    "boolean",
+    "cat",
+    "corr_cholesky",
+    "dependent",
+    "dependent_property",
+    "greater_than",
+    "greater_than_eq",
+    "independent",
+    "integer_interval",
+    "interval",
+    "half_open_interval",
+    "is_dependent",
+    "less_than",
+    "lower_cholesky",
+    "lower_triangular",
+    "multinomial",
+    "nonnegative",
+    "nonnegative_integer",
+    "one_hot",
+    "positive",
+    "positive_semidefinite",
+    "positive_definite",
+    "positive_integer",
+    "real",
+    "real_vector",
+    "simplex",
+    "square",
+    "stack",
+    "symmetric",
+    "unit_interval",
+]
+
+
+class Constraint:
+    """
+    Abstract base class for constraints.
+
+    A constraint object represents a region over which a variable is valid,
+    e.g. within which a variable can be optimized.
+
+    Attributes:
+        is_discrete (bool): Whether constrained space is discrete.
+            Defaults to False.
+        event_dim (int): Number of rightmost dimensions that together define
+            an event. The :meth:`check` method will remove this many dimensions
+            when computing validity.
+    """
+
+    is_discrete = False  # Default to continuous.
+    event_dim = 0  # Default to univariate.
+
+    def check(self, value):
+        """
+        Returns a byte tensor of ``sample_shape + batch_shape`` indicating
+        whether each event in value satisfies this constraint.
+        """
+        raise NotImplementedError
+
+    def __repr__(self):
+        return self.__class__.__name__[1:] + "()"
+
+
+class _Dependent(Constraint):
+    """
+    Placeholder for variables whose support depends on other variables.
+    These variables obey no simple coordinate-wise constraints.
+
+    Args:
+        is_discrete (bool): Optional value of ``.is_discrete`` in case this
+            can be computed statically. If not provided, access to the
+            ``.is_discrete`` attribute will raise a NotImplementedError.
+        event_dim (int): Optional value of ``.event_dim`` in case this
+            can be computed statically. If not provided, access to the
+            ``.event_dim`` attribute will raise a NotImplementedError.
+    """
+
+    def __init__(self, *, is_discrete=NotImplemented, event_dim=NotImplemented):
+        self._is_discrete = is_discrete
+        self._event_dim = event_dim
+        super().__init__()
+
+    @property
+    def is_discrete(self):
+        if self._is_discrete is NotImplemented:
+            raise NotImplementedError(".is_discrete cannot be determined statically")
+        return self._is_discrete
+
+    @property
+    def event_dim(self):
+        if self._event_dim is NotImplemented:
+            raise NotImplementedError(".event_dim cannot be determined statically")
+        return self._event_dim
+
+    def __call__(self, *, is_discrete=NotImplemented, event_dim=NotImplemented):
+        """
+        Support for syntax to customize static attributes::
+
+            constraints.dependent(is_discrete=True, event_dim=1)
+        """
+        if is_discrete is NotImplemented:
+            is_discrete = self._is_discrete
+        if event_dim is NotImplemented:
+            event_dim = self._event_dim
+        return _Dependent(is_discrete=is_discrete, event_dim=event_dim)
+
+    def check(self, x):
+        raise ValueError("Cannot determine validity of dependent constraint")
+
+
+def is_dependent(constraint):
+    """
+    Checks if ``constraint`` is a ``_Dependent`` object.
+
+    Args:
+        constraint : A ``Constraint`` object.
+
+    Returns:
+        ``bool``: True if ``constraint`` can be refined to the type ``_Dependent``, False otherwise.
+
+    Examples:
+        >>> import torch
+        >>> from torch.distributions import Bernoulli
+        >>> from torch.distributions.constraints import is_dependent
+
+        >>> dist = Bernoulli(probs = torch.tensor([0.6], requires_grad=True))
+        >>> constraint1 = dist.arg_constraints["probs"]
+        >>> constraint2 = dist.arg_constraints["logits"]
+
+        >>> for constraint in [constraint1, constraint2]:
+        >>>     if is_dependent(constraint):
+        >>>         continue
+    """
+    return isinstance(constraint, _Dependent)
+
+
+class _DependentProperty(property, _Dependent):
+    """
+    Decorator that extends @property to act like a `Dependent` constraint when
+    called on a class and act like a property when called on an object.
+
+    Example::
+
+        class Uniform(Distribution):
+            def __init__(self, low, high):
+                self.low = low
+                self.high = high
+            @constraints.dependent_property(is_discrete=False, event_dim=0)
+            def support(self):
+                return constraints.interval(self.low, self.high)
+
+    Args:
+        fn (Callable): The function to be decorated.
+        is_discrete (bool): Optional value of ``.is_discrete`` in case this
+            can be computed statically. If not provided, access to the
+            ``.is_discrete`` attribute will raise a NotImplementedError.
+        event_dim (int): Optional value of ``.event_dim`` in case this
+            can be computed statically. If not provided, access to the
+            ``.event_dim`` attribute will raise a NotImplementedError.
+    """
+
+    def __init__(
+        self, fn=None, *, is_discrete=NotImplemented, event_dim=NotImplemented
+    ):
+        super().__init__(fn)
+        self._is_discrete = is_discrete
+        self._event_dim = event_dim
+
+    def __call__(self, fn):
+        """
+        Support for syntax to customize static attributes::
+
+            @constraints.dependent_property(is_discrete=True, event_dim=1)
+            def support(self):
+                ...
+        """
+        return _DependentProperty(
+            fn, is_discrete=self._is_discrete, event_dim=self._event_dim
+        )
+
+
+class _IndependentConstraint(Constraint):
+    """
+    Wraps a constraint by aggregating over ``reinterpreted_batch_ndims``-many
+    dims in :meth:`check`, so that an event is valid only if all its
+    independent entries are valid.
+    """
+
+    def __init__(self, base_constraint, reinterpreted_batch_ndims):
+        assert isinstance(base_constraint, Constraint)
+        assert isinstance(reinterpreted_batch_ndims, int)
+        assert reinterpreted_batch_ndims >= 0
+        self.base_constraint = base_constraint
+        self.reinterpreted_batch_ndims = reinterpreted_batch_ndims
+        super().__init__()
+
+    @property
+    def is_discrete(self):
+        return self.base_constraint.is_discrete
+
+    @property
+    def event_dim(self):
+        return self.base_constraint.event_dim + self.reinterpreted_batch_ndims
+
+    def check(self, value):
+        result = self.base_constraint.check(value)
+        if result.dim() < self.reinterpreted_batch_ndims:
+            expected = self.base_constraint.event_dim + self.reinterpreted_batch_ndims
+            raise ValueError(
+                f"Expected value.dim() >= {expected} but got {value.dim()}"
+            )
+        result = result.reshape(
+            result.shape[: result.dim() - self.reinterpreted_batch_ndims] + (-1,)
+        )
+        result = result.all(-1)
+        return result
+
+    def __repr__(self):
+        return f"{self.__class__.__name__[1:]}({repr(self.base_constraint)}, {self.reinterpreted_batch_ndims})"
+
+
+class _Boolean(Constraint):
+    """
+    Constrain to the two values `{0, 1}`.
+    """
+
+    is_discrete = True
+
+    def check(self, value):
+        return (value == 0) | (value == 1)
+
+
+class _OneHot(Constraint):
+    """
+    Constrain to one-hot vectors.
+    """
+
+    is_discrete = True
+    event_dim = 1
+
+    def check(self, value):
+        is_boolean = (value == 0) | (value == 1)
+        is_normalized = value.sum(-1).eq(1)
+        return is_boolean.all(-1) & is_normalized
+
+
+class _IntegerInterval(Constraint):
+    """
+    Constrain to an integer interval `[lower_bound, upper_bound]`.
+    """
+
+    is_discrete = True
+
+    def __init__(self, lower_bound, upper_bound):
+        self.lower_bound = lower_bound
+        self.upper_bound = upper_bound
+        super().__init__()
+
+    def check(self, value):
+        return (
+            (value % 1 == 0) & (self.lower_bound <= value) & (value <= self.upper_bound)
+        )
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += (
+            f"(lower_bound={self.lower_bound}, upper_bound={self.upper_bound})"
+        )
+        return fmt_string
+
+
+class _IntegerLessThan(Constraint):
+    """
+    Constrain to an integer interval `(-inf, upper_bound]`.
+    """
+
+    is_discrete = True
+
+    def __init__(self, upper_bound):
+        self.upper_bound = upper_bound
+        super().__init__()
+
+    def check(self, value):
+        return (value % 1 == 0) & (value <= self.upper_bound)
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += f"(upper_bound={self.upper_bound})"
+        return fmt_string
+
+
+class _IntegerGreaterThan(Constraint):
+    """
+    Constrain to an integer interval `[lower_bound, inf)`.
+    """
+
+    is_discrete = True
+
+    def __init__(self, lower_bound):
+        self.lower_bound = lower_bound
+        super().__init__()
+
+    def check(self, value):
+        return (value % 1 == 0) & (value >= self.lower_bound)
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += f"(lower_bound={self.lower_bound})"
+        return fmt_string
+
+
+class _Real(Constraint):
+    """
+    Trivially constrain to the extended real line `[-inf, inf]`.
+    """
+
+    def check(self, value):
+        return value == value  # False for NANs.
+
+
+class _GreaterThan(Constraint):
+    """
+    Constrain to a real half line `(lower_bound, inf]`.
+    """
+
+    def __init__(self, lower_bound):
+        self.lower_bound = lower_bound
+        super().__init__()
+
+    def check(self, value):
+        return self.lower_bound < value
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += f"(lower_bound={self.lower_bound})"
+        return fmt_string
+
+
+class _GreaterThanEq(Constraint):
+    """
+    Constrain to a real half line `[lower_bound, inf)`.
+    """
+
+    def __init__(self, lower_bound):
+        self.lower_bound = lower_bound
+        super().__init__()
+
+    def check(self, value):
+        return self.lower_bound <= value
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += f"(lower_bound={self.lower_bound})"
+        return fmt_string
+
+
+class _LessThan(Constraint):
+    """
+    Constrain to a real half line `[-inf, upper_bound)`.
+    """
+
+    def __init__(self, upper_bound):
+        self.upper_bound = upper_bound
+        super().__init__()
+
+    def check(self, value):
+        return value < self.upper_bound
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += f"(upper_bound={self.upper_bound})"
+        return fmt_string
+
+
+class _Interval(Constraint):
+    """
+    Constrain to a real interval `[lower_bound, upper_bound]`.
+    """
+
+    def __init__(self, lower_bound, upper_bound):
+        self.lower_bound = lower_bound
+        self.upper_bound = upper_bound
+        super().__init__()
+
+    def check(self, value):
+        return (self.lower_bound <= value) & (value <= self.upper_bound)
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += (
+            f"(lower_bound={self.lower_bound}, upper_bound={self.upper_bound})"
+        )
+        return fmt_string
+
+
+class _HalfOpenInterval(Constraint):
+    """
+    Constrain to a real interval `[lower_bound, upper_bound)`.
+    """
+
+    def __init__(self, lower_bound, upper_bound):
+        self.lower_bound = lower_bound
+        self.upper_bound = upper_bound
+        super().__init__()
+
+    def check(self, value):
+        return (self.lower_bound <= value) & (value < self.upper_bound)
+
+    def __repr__(self):
+        fmt_string = self.__class__.__name__[1:]
+        fmt_string += (
+            f"(lower_bound={self.lower_bound}, upper_bound={self.upper_bound})"
+        )
+        return fmt_string
+
+
+class _Simplex(Constraint):
+    """
+    Constrain to the unit simplex in the innermost (rightmost) dimension.
+    Specifically: `x >= 0` and `x.sum(-1) == 1`.
+    """
+
+    event_dim = 1
+
+    def check(self, value):
+        return torch.all(value >= 0, dim=-1) & ((value.sum(-1) - 1).abs() < 1e-6)
+
+
+class _Multinomial(Constraint):
+    """
+    Constrain to nonnegative integer values summing to at most an upper bound.
+
+    Note due to limitations of the Multinomial distribution, this currently
+    checks the weaker condition ``value.sum(-1) <= upper_bound``. In the future
+    this may be strengthened to ``value.sum(-1) == upper_bound``.
+    """
+
+    is_discrete = True
+    event_dim = 1
+
+    def __init__(self, upper_bound):
+        self.upper_bound = upper_bound
+
+    def check(self, x):
+        return (x >= 0).all(dim=-1) & (x.sum(dim=-1) <= self.upper_bound)
+
+
+class _LowerTriangular(Constraint):
+    """
+    Constrain to lower-triangular square matrices.
+    """
+
+    event_dim = 2
+
+    def check(self, value):
+        value_tril = value.tril()
+        return (value_tril == value).view(value.shape[:-2] + (-1,)).min(-1)[0]
+
+
+class _LowerCholesky(Constraint):
+    """
+    Constrain to lower-triangular square matrices with positive diagonals.
+    """
+
+    event_dim = 2
+
+    def check(self, value):
+        value_tril = value.tril()
+        lower_triangular = (
+            (value_tril == value).view(value.shape[:-2] + (-1,)).min(-1)[0]
+        )
+
+        positive_diagonal = (value.diagonal(dim1=-2, dim2=-1) > 0).min(-1)[0]
+        return lower_triangular & positive_diagonal
+
+
+class _CorrCholesky(Constraint):
+    """
+    Constrain to lower-triangular square matrices with positive diagonals and each
+    row vector being of unit length.
+    """
+
+    event_dim = 2
+
+    def check(self, value):
+        tol = (
+            torch.finfo(value.dtype).eps * value.size(-1) * 10
+        )  # 10 is an adjustable fudge factor
+        row_norm = torch.linalg.norm(value.detach(), dim=-1)
+        unit_row_norm = (row_norm - 1.0).abs().le(tol).all(dim=-1)
+        return _LowerCholesky().check(value) & unit_row_norm
+
+
+class _Square(Constraint):
+    """
+    Constrain to square matrices.
+    """
+
+    event_dim = 2
+
+    def check(self, value):
+        return torch.full(
+            size=value.shape[:-2],
+            fill_value=(value.shape[-2] == value.shape[-1]),
+            dtype=torch.bool,
+            device=value.device,
+        )
+
+
+class _Symmetric(_Square):
+    """
+    Constrain to Symmetric square matrices.
+    """
+
+    def check(self, value):
+        square_check = super().check(value)
+        if not square_check.all():
+            return square_check
+        return torch.isclose(value, value.mT, atol=1e-6).all(-2).all(-1)
+
+
+class _PositiveSemidefinite(_Symmetric):
+    """
+    Constrain to positive-semidefinite matrices.
+    """
+
+    def check(self, value):
+        sym_check = super().check(value)
+        if not sym_check.all():
+            return sym_check
+        return torch.linalg.eigvalsh(value).ge(0).all(-1)
+
+
+class _PositiveDefinite(_Symmetric):
+    """
+    Constrain to positive-definite matrices.
+    """
+
+    def check(self, value):
+        sym_check = super().check(value)
+        if not sym_check.all():
+            return sym_check
+        return torch.linalg.cholesky_ex(value).info.eq(0)
+
+
+class _Cat(Constraint):
+    """
+    Constraint functor that applies a sequence of constraints
+    `cseq` at the submatrices at dimension `dim`,
+    each of size `lengths[dim]`, in a way compatible with :func:`torch.cat`.
+    """
+
+    def __init__(self, cseq, dim=0, lengths=None):
+        assert all(isinstance(c, Constraint) for c in cseq)
+        self.cseq = list(cseq)
+        if lengths is None:
+            lengths = [1] * len(self.cseq)
+        self.lengths = list(lengths)
+        assert len(self.lengths) == len(self.cseq)
+        self.dim = dim
+        super().__init__()
+
+    @property
+    def is_discrete(self):
+        return any(c.is_discrete for c in self.cseq)
+
+    @property
+    def event_dim(self):
+        return max(c.event_dim for c in self.cseq)
+
+    def check(self, value):
+        assert -value.dim() <= self.dim < value.dim()
+        checks = []
+        start = 0
+        for constr, length in zip(self.cseq, self.lengths):
+            v = value.narrow(self.dim, start, length)
+            checks.append(constr.check(v))
+            start = start + length  # avoid += for jit compat
+        return torch.cat(checks, self.dim)
+
+
+class _Stack(Constraint):
+    """
+    Constraint functor that applies a sequence of constraints
+    `cseq` at the submatrices at dimension `dim`,
+    in a way compatible with :func:`torch.stack`.
+    """
+
+    def __init__(self, cseq, dim=0):
+        assert all(isinstance(c, Constraint) for c in cseq)
+        self.cseq = list(cseq)
+        self.dim = dim
+        super().__init__()
+
+    @property
+    def is_discrete(self):
+        return any(c.is_discrete for c in self.cseq)
+
+    @property
+    def event_dim(self):
+        dim = max(c.event_dim for c in self.cseq)
+        if self.dim + dim < 0:
+            dim += 1
+        return dim
+
+    def check(self, value):
+        assert -value.dim() <= self.dim < value.dim()
+        vs = [value.select(self.dim, i) for i in range(value.size(self.dim))]
+        return torch.stack(
+            [constr.check(v) for v, constr in zip(vs, self.cseq)], self.dim
+        )
+
+
+# Public interface.
+dependent = _Dependent()
+dependent_property = _DependentProperty
+independent = _IndependentConstraint
+boolean = _Boolean()
+one_hot = _OneHot()
+nonnegative_integer = _IntegerGreaterThan(0)
+positive_integer = _IntegerGreaterThan(1)
+integer_interval = _IntegerInterval
+real = _Real()
+real_vector = independent(real, 1)
+positive = _GreaterThan(0.0)
+nonnegative = _GreaterThanEq(0.0)
+greater_than = _GreaterThan
+greater_than_eq = _GreaterThanEq
+less_than = _LessThan
+multinomial = _Multinomial
+unit_interval = _Interval(0.0, 1.0)
+interval = _Interval
+half_open_interval = _HalfOpenInterval
+simplex = _Simplex()
+lower_triangular = _LowerTriangular()
+lower_cholesky = _LowerCholesky()
+corr_cholesky = _CorrCholesky()
+square = _Square()
+symmetric = _Symmetric()
+positive_semidefinite = _PositiveSemidefinite()
+positive_definite = _PositiveDefinite()
+cat = _Cat
+stack = _Stack

rl/Lib/site-packages/torch/distributions/continuous_bernoulli.py

@@ -0,0 +1,238 @@
+# mypy: allow-untyped-defs
+import math
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import (
+    broadcast_all,
+    clamp_probs,
+    lazy_property,
+    logits_to_probs,
+    probs_to_logits,
+)
+from torch.nn.functional import binary_cross_entropy_with_logits
+from torch.types import _size
+
+
+__all__ = ["ContinuousBernoulli"]
+
+
+class ContinuousBernoulli(ExponentialFamily):
+    r"""
+    Creates a continuous Bernoulli distribution parameterized by :attr:`probs`
+    or :attr:`logits` (but not both).
+
+    The distribution is supported in [0, 1] and parameterized by 'probs' (in
+    (0,1)) or 'logits' (real-valued). Note that, unlike the Bernoulli, 'probs'
+    does not correspond to a probability and 'logits' does not correspond to
+    log-odds, but the same names are used due to the similarity with the
+    Bernoulli. See [1] for more details.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = ContinuousBernoulli(torch.tensor([0.3]))
+        >>> m.sample()
+        tensor([ 0.2538])
+
+    Args:
+        probs (Number, Tensor): (0,1) valued parameters
+        logits (Number, Tensor): real valued parameters whose sigmoid matches 'probs'
+
+    [1] The continuous Bernoulli: fixing a pervasive error in variational
+    autoencoders, Loaiza-Ganem G and Cunningham JP, NeurIPS 2019.
+    https://arxiv.org/abs/1907.06845
+    """
+    arg_constraints = {"probs": constraints.unit_interval, "logits": constraints.real}
+    support = constraints.unit_interval
+    _mean_carrier_measure = 0
+    has_rsample = True
+
+    def __init__(
+        self, probs=None, logits=None, lims=(0.499, 0.501), validate_args=None
+    ):
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            is_scalar = isinstance(probs, Number)
+            (self.probs,) = broadcast_all(probs)
+            # validate 'probs' here if necessary as it is later clamped for numerical stability
+            # close to 0 and 1, later on; otherwise the clamped 'probs' would always pass
+            if validate_args is not None:
+                if not self.arg_constraints["probs"].check(self.probs).all():
+                    raise ValueError("The parameter probs has invalid values")
+            self.probs = clamp_probs(self.probs)
+        else:
+            is_scalar = isinstance(logits, Number)
+            (self.logits,) = broadcast_all(logits)
+        self._param = self.probs if probs is not None else self.logits
+        if is_scalar:
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self._param.size()
+        self._lims = lims
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(ContinuousBernoulli, _instance)
+        new._lims = self._lims
+        batch_shape = torch.Size(batch_shape)
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(batch_shape)
+            new._param = new.probs
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(batch_shape)
+            new._param = new.logits
+        super(ContinuousBernoulli, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._param.new(*args, **kwargs)
+
+    def _outside_unstable_region(self):
+        return torch.max(
+            torch.le(self.probs, self._lims[0]), torch.gt(self.probs, self._lims[1])
+        )
+
+    def _cut_probs(self):
+        return torch.where(
+            self._outside_unstable_region(),
+            self.probs,
+            self._lims[0] * torch.ones_like(self.probs),
+        )
+
+    def _cont_bern_log_norm(self):
+        """computes the log normalizing constant as a function of the 'probs' parameter"""
+        cut_probs = self._cut_probs()
+        cut_probs_below_half = torch.where(
+            torch.le(cut_probs, 0.5), cut_probs, torch.zeros_like(cut_probs)
+        )
+        cut_probs_above_half = torch.where(
+            torch.ge(cut_probs, 0.5), cut_probs, torch.ones_like(cut_probs)
+        )
+        log_norm = torch.log(
+            torch.abs(torch.log1p(-cut_probs) - torch.log(cut_probs))
+        ) - torch.where(
+            torch.le(cut_probs, 0.5),
+            torch.log1p(-2.0 * cut_probs_below_half),
+            torch.log(2.0 * cut_probs_above_half - 1.0),
+        )
+        x = torch.pow(self.probs - 0.5, 2)
+        taylor = math.log(2.0) + (4.0 / 3.0 + 104.0 / 45.0 * x) * x
+        return torch.where(self._outside_unstable_region(), log_norm, taylor)
+
+    @property
+    def mean(self):
+        cut_probs = self._cut_probs()
+        mus = cut_probs / (2.0 * cut_probs - 1.0) + 1.0 / (
+            torch.log1p(-cut_probs) - torch.log(cut_probs)
+        )
+        x = self.probs - 0.5
+        taylor = 0.5 + (1.0 / 3.0 + 16.0 / 45.0 * torch.pow(x, 2)) * x
+        return torch.where(self._outside_unstable_region(), mus, taylor)
+
+    @property
+    def stddev(self):
+        return torch.sqrt(self.variance)
+
+    @property
+    def variance(self):
+        cut_probs = self._cut_probs()
+        vars = cut_probs * (cut_probs - 1.0) / torch.pow(
+            1.0 - 2.0 * cut_probs, 2
+        ) + 1.0 / torch.pow(torch.log1p(-cut_probs) - torch.log(cut_probs), 2)
+        x = torch.pow(self.probs - 0.5, 2)
+        taylor = 1.0 / 12.0 - (1.0 / 15.0 - 128.0 / 945.0 * x) * x
+        return torch.where(self._outside_unstable_region(), vars, taylor)
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs, is_binary=True)
+
+    @lazy_property
+    def probs(self):
+        return clamp_probs(logits_to_probs(self.logits, is_binary=True))
+
+    @property
+    def param_shape(self):
+        return self._param.size()
+
+    def sample(self, sample_shape=torch.Size()):
+        shape = self._extended_shape(sample_shape)
+        u = torch.rand(shape, dtype=self.probs.dtype, device=self.probs.device)
+        with torch.no_grad():
+            return self.icdf(u)
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        u = torch.rand(shape, dtype=self.probs.dtype, device=self.probs.device)
+        return self.icdf(u)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        logits, value = broadcast_all(self.logits, value)
+        return (
+            -binary_cross_entropy_with_logits(logits, value, reduction="none")
+            + self._cont_bern_log_norm()
+        )
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        cut_probs = self._cut_probs()
+        cdfs = (
+            torch.pow(cut_probs, value) * torch.pow(1.0 - cut_probs, 1.0 - value)
+            + cut_probs
+            - 1.0
+        ) / (2.0 * cut_probs - 1.0)
+        unbounded_cdfs = torch.where(self._outside_unstable_region(), cdfs, value)
+        return torch.where(
+            torch.le(value, 0.0),
+            torch.zeros_like(value),
+            torch.where(torch.ge(value, 1.0), torch.ones_like(value), unbounded_cdfs),
+        )
+
+    def icdf(self, value):
+        cut_probs = self._cut_probs()
+        return torch.where(
+            self._outside_unstable_region(),
+            (
+                torch.log1p(-cut_probs + value * (2.0 * cut_probs - 1.0))
+                - torch.log1p(-cut_probs)
+            )
+            / (torch.log(cut_probs) - torch.log1p(-cut_probs)),
+            value,
+        )
+
+    def entropy(self):
+        log_probs0 = torch.log1p(-self.probs)
+        log_probs1 = torch.log(self.probs)
+        return (
+            self.mean * (log_probs0 - log_probs1)
+            - self._cont_bern_log_norm()
+            - log_probs0
+        )
+
+    @property
+    def _natural_params(self):
+        return (self.logits,)
+
+    def _log_normalizer(self, x):
+        """computes the log normalizing constant as a function of the natural parameter"""
+        out_unst_reg = torch.max(
+            torch.le(x, self._lims[0] - 0.5), torch.gt(x, self._lims[1] - 0.5)
+        )
+        cut_nat_params = torch.where(
+            out_unst_reg, x, (self._lims[0] - 0.5) * torch.ones_like(x)
+        )
+        log_norm = torch.log(torch.abs(torch.exp(cut_nat_params) - 1.0)) - torch.log(
+            torch.abs(cut_nat_params)
+        )
+        taylor = 0.5 * x + torch.pow(x, 2) / 24.0 - torch.pow(x, 4) / 2880.0
+        return torch.where(out_unst_reg, log_norm, taylor)

rl/Lib/site-packages/torch/distributions/dirichlet.py

@@ -0,0 +1,126 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.autograd import Function
+from torch.autograd.function import once_differentiable
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.types import _size
+
+
+__all__ = ["Dirichlet"]
+
+
+# This helper is exposed for testing.
+def _Dirichlet_backward(x, concentration, grad_output):
+    total = concentration.sum(-1, True).expand_as(concentration)
+    grad = torch._dirichlet_grad(x, concentration, total)
+    return grad * (grad_output - (x * grad_output).sum(-1, True))
+
+
+class _Dirichlet(Function):
+    @staticmethod
+    def forward(ctx, concentration):
+        x = torch._sample_dirichlet(concentration)
+        ctx.save_for_backward(x, concentration)
+        return x
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output):
+        x, concentration = ctx.saved_tensors
+        return _Dirichlet_backward(x, concentration, grad_output)
+
+
+class Dirichlet(ExponentialFamily):
+    r"""
+    Creates a Dirichlet distribution parameterized by concentration :attr:`concentration`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Dirichlet(torch.tensor([0.5, 0.5]))
+        >>> m.sample()  # Dirichlet distributed with concentration [0.5, 0.5]
+        tensor([ 0.1046,  0.8954])
+
+    Args:
+        concentration (Tensor): concentration parameter of the distribution
+            (often referred to as alpha)
+    """
+    arg_constraints = {
+        "concentration": constraints.independent(constraints.positive, 1)
+    }
+    support = constraints.simplex
+    has_rsample = True
+
+    def __init__(self, concentration, validate_args=None):
+        if concentration.dim() < 1:
+            raise ValueError(
+                "`concentration` parameter must be at least one-dimensional."
+            )
+        self.concentration = concentration
+        batch_shape, event_shape = concentration.shape[:-1], concentration.shape[-1:]
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Dirichlet, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.concentration = self.concentration.expand(batch_shape + self.event_shape)
+        super(Dirichlet, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    def rsample(self, sample_shape: _size = ()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        concentration = self.concentration.expand(shape)
+        return _Dirichlet.apply(concentration)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return (
+            torch.xlogy(self.concentration - 1.0, value).sum(-1)
+            + torch.lgamma(self.concentration.sum(-1))
+            - torch.lgamma(self.concentration).sum(-1)
+        )
+
+    @property
+    def mean(self):
+        return self.concentration / self.concentration.sum(-1, True)
+
+    @property
+    def mode(self):
+        concentrationm1 = (self.concentration - 1).clamp(min=0.0)
+        mode = concentrationm1 / concentrationm1.sum(-1, True)
+        mask = (self.concentration < 1).all(axis=-1)
+        mode[mask] = torch.nn.functional.one_hot(
+            mode[mask].argmax(axis=-1), concentrationm1.shape[-1]
+        ).to(mode)
+        return mode
+
+    @property
+    def variance(self):
+        con0 = self.concentration.sum(-1, True)
+        return (
+            self.concentration
+            * (con0 - self.concentration)
+            / (con0.pow(2) * (con0 + 1))
+        )
+
+    def entropy(self):
+        k = self.concentration.size(-1)
+        a0 = self.concentration.sum(-1)
+        return (
+            torch.lgamma(self.concentration).sum(-1)
+            - torch.lgamma(a0)
+            - (k - a0) * torch.digamma(a0)
+            - ((self.concentration - 1.0) * torch.digamma(self.concentration)).sum(-1)
+        )
+
+    @property
+    def _natural_params(self):
+        return (self.concentration,)
+
+    def _log_normalizer(self, x):
+        return x.lgamma().sum(-1) - torch.lgamma(x.sum(-1))

rl/Lib/site-packages/torch/distributions/distribution.py

@@ -0,0 +1,340 @@
+# mypy: allow-untyped-defs
+import warnings
+from typing import Any, Dict, Optional
+from typing_extensions import deprecated
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.utils import lazy_property
+from torch.types import _size
+
+
+__all__ = ["Distribution"]
+
+
+class Distribution:
+    r"""
+    Distribution is the abstract base class for probability distributions.
+    """
+
+    has_rsample = False
+    has_enumerate_support = False
+    _validate_args = __debug__
+
+    @staticmethod
+    def set_default_validate_args(value: bool) -> None:
+        """
+        Sets whether validation is enabled or disabled.
+
+        The default behavior mimics Python's ``assert`` statement: validation
+        is on by default, but is disabled if Python is run in optimized mode
+        (via ``python -O``). Validation may be expensive, so you may want to
+        disable it once a model is working.
+
+        Args:
+            value (bool): Whether to enable validation.
+        """
+        if value not in [True, False]:
+            raise ValueError
+        Distribution._validate_args = value
+
+    def __init__(
+        self,
+        batch_shape: torch.Size = torch.Size(),
+        event_shape: torch.Size = torch.Size(),
+        validate_args: Optional[bool] = None,
+    ):
+        self._batch_shape = batch_shape
+        self._event_shape = event_shape
+        if validate_args is not None:
+            self._validate_args = validate_args
+        if self._validate_args:
+            try:
+                arg_constraints = self.arg_constraints
+            except NotImplementedError:
+                arg_constraints = {}
+                warnings.warn(
+                    f"{self.__class__} does not define `arg_constraints`. "
+                    + "Please set `arg_constraints = {}` or initialize the distribution "
+                    + "with `validate_args=False` to turn off validation."
+                )
+            for param, constraint in arg_constraints.items():
+                if constraints.is_dependent(constraint):
+                    continue  # skip constraints that cannot be checked
+                if param not in self.__dict__ and isinstance(
+                    getattr(type(self), param), lazy_property
+                ):
+                    continue  # skip checking lazily-constructed args
+                value = getattr(self, param)
+                valid = constraint.check(value)
+                if not valid.all():
+                    raise ValueError(
+                        f"Expected parameter {param} "
+                        f"({type(value).__name__} of shape {tuple(value.shape)}) "
+                        f"of distribution {repr(self)} "
+                        f"to satisfy the constraint {repr(constraint)}, "
+                        f"but found invalid values:\n{value}"
+                    )
+        super().__init__()
+
+    def expand(self, batch_shape: _size, _instance=None):
+        """
+        Returns a new distribution instance (or populates an existing instance
+        provided by a derived class) with batch dimensions expanded to
+        `batch_shape`. This method calls :class:`~torch.Tensor.expand` on
+        the distribution's parameters. As such, this does not allocate new
+        memory for the expanded distribution instance. Additionally,
+        this does not repeat any args checking or parameter broadcasting in
+        `__init__.py`, when an instance is first created.
+
+        Args:
+            batch_shape (torch.Size): the desired expanded size.
+            _instance: new instance provided by subclasses that
+                need to override `.expand`.
+
+        Returns:
+            New distribution instance with batch dimensions expanded to
+            `batch_size`.
+        """
+        raise NotImplementedError
+
+    @property
+    def batch_shape(self) -> torch.Size:
+        """
+        Returns the shape over which parameters are batched.
+        """
+        return self._batch_shape
+
+    @property
+    def event_shape(self) -> torch.Size:
+        """
+        Returns the shape of a single sample (without batching).
+        """
+        return self._event_shape
+
+    @property
+    def arg_constraints(self) -> Dict[str, constraints.Constraint]:
+        """
+        Returns a dictionary from argument names to
+        :class:`~torch.distributions.constraints.Constraint` objects that
+        should be satisfied by each argument of this distribution. Args that
+        are not tensors need not appear in this dict.
+        """
+        raise NotImplementedError
+
+    @property
+    def support(self) -> Optional[Any]:
+        """
+        Returns a :class:`~torch.distributions.constraints.Constraint` object
+        representing this distribution's support.
+        """
+        raise NotImplementedError
+
+    @property
+    def mean(self) -> torch.Tensor:
+        """
+        Returns the mean of the distribution.
+        """
+        raise NotImplementedError
+
+    @property
+    def mode(self) -> torch.Tensor:
+        """
+        Returns the mode of the distribution.
+        """
+        raise NotImplementedError(f"{self.__class__} does not implement mode")
+
+    @property
+    def variance(self) -> torch.Tensor:
+        """
+        Returns the variance of the distribution.
+        """
+        raise NotImplementedError
+
+    @property
+    def stddev(self) -> torch.Tensor:
+        """
+        Returns the standard deviation of the distribution.
+        """
+        return self.variance.sqrt()
+
+    def sample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        """
+        Generates a sample_shape shaped sample or sample_shape shaped batch of
+        samples if the distribution parameters are batched.
+        """
+        with torch.no_grad():
+            return self.rsample(sample_shape)
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        """
+        Generates a sample_shape shaped reparameterized sample or sample_shape
+        shaped batch of reparameterized samples if the distribution parameters
+        are batched.
+        """
+        raise NotImplementedError
+
+    @deprecated(
+        "`sample_n(n)` will be deprecated. Use `sample((n,))` instead.",
+        category=FutureWarning,
+    )
+    def sample_n(self, n: int) -> torch.Tensor:
+        """
+        Generates n samples or n batches of samples if the distribution
+        parameters are batched.
+        """
+        return self.sample(torch.Size((n,)))
+
+    def log_prob(self, value: torch.Tensor) -> torch.Tensor:
+        """
+        Returns the log of the probability density/mass function evaluated at
+        `value`.
+
+        Args:
+            value (Tensor):
+        """
+        raise NotImplementedError
+
+    def cdf(self, value: torch.Tensor) -> torch.Tensor:
+        """
+        Returns the cumulative density/mass function evaluated at
+        `value`.
+
+        Args:
+            value (Tensor):
+        """
+        raise NotImplementedError
+
+    def icdf(self, value: torch.Tensor) -> torch.Tensor:
+        """
+        Returns the inverse cumulative density/mass function evaluated at
+        `value`.
+
+        Args:
+            value (Tensor):
+        """
+        raise NotImplementedError
+
+    def enumerate_support(self, expand: bool = True) -> torch.Tensor:
+        """
+        Returns tensor containing all values supported by a discrete
+        distribution. The result will enumerate over dimension 0, so the shape
+        of the result will be `(cardinality,) + batch_shape + event_shape`
+        (where `event_shape = ()` for univariate distributions).
+
+        Note that this enumerates over all batched tensors in lock-step
+        `[[0, 0], [1, 1], ...]`. With `expand=False`, enumeration happens
+        along dim 0, but with the remaining batch dimensions being
+        singleton dimensions, `[[0], [1], ..`.
+
+        To iterate over the full Cartesian product use
+        `itertools.product(m.enumerate_support())`.
+
+        Args:
+            expand (bool): whether to expand the support over the
+                batch dims to match the distribution's `batch_shape`.
+
+        Returns:
+            Tensor iterating over dimension 0.
+        """
+        raise NotImplementedError
+
+    def entropy(self) -> torch.Tensor:
+        """
+        Returns entropy of distribution, batched over batch_shape.
+
+        Returns:
+            Tensor of shape batch_shape.
+        """
+        raise NotImplementedError
+
+    def perplexity(self) -> torch.Tensor:
+        """
+        Returns perplexity of distribution, batched over batch_shape.
+
+        Returns:
+            Tensor of shape batch_shape.
+        """
+        return torch.exp(self.entropy())
+
+    def _extended_shape(self, sample_shape: _size = torch.Size()) -> torch.Size:
+        """
+        Returns the size of the sample returned by the distribution, given
+        a `sample_shape`. Note, that the batch and event shapes of a distribution
+        instance are fixed at the time of construction. If this is empty, the
+        returned shape is upcast to (1,).
+
+        Args:
+            sample_shape (torch.Size): the size of the sample to be drawn.
+        """
+        if not isinstance(sample_shape, torch.Size):
+            sample_shape = torch.Size(sample_shape)
+        return torch.Size(sample_shape + self._batch_shape + self._event_shape)
+
+    def _validate_sample(self, value: torch.Tensor) -> None:
+        """
+        Argument validation for distribution methods such as `log_prob`,
+        `cdf` and `icdf`. The rightmost dimensions of a value to be
+        scored via these methods must agree with the distribution's batch
+        and event shapes.
+
+        Args:
+            value (Tensor): the tensor whose log probability is to be
+                computed by the `log_prob` method.
+        Raises
+            ValueError: when the rightmost dimensions of `value` do not match the
+                distribution's batch and event shapes.
+        """
+        if not isinstance(value, torch.Tensor):
+            raise ValueError("The value argument to log_prob must be a Tensor")
+
+        event_dim_start = len(value.size()) - len(self._event_shape)
+        if value.size()[event_dim_start:] != self._event_shape:
+            raise ValueError(
+                f"The right-most size of value must match event_shape: {value.size()} vs {self._event_shape}."
+            )
+
+        actual_shape = value.size()
+        expected_shape = self._batch_shape + self._event_shape
+        for i, j in zip(reversed(actual_shape), reversed(expected_shape)):
+            if i != 1 and j != 1 and i != j:
+                raise ValueError(
+                    f"Value is not broadcastable with batch_shape+event_shape: {actual_shape} vs {expected_shape}."
+                )
+        try:
+            support = self.support
+        except NotImplementedError:
+            warnings.warn(
+                f"{self.__class__} does not define `support` to enable "
+                + "sample validation. Please initialize the distribution with "
+                + "`validate_args=False` to turn off validation."
+            )
+            return
+        assert support is not None
+        valid = support.check(value)
+        if not valid.all():
+            raise ValueError(
+                "Expected value argument "
+                f"({type(value).__name__} of shape {tuple(value.shape)}) "
+                f"to be within the support ({repr(support)}) "
+                f"of the distribution {repr(self)}, "
+                f"but found invalid values:\n{value}"
+            )
+
+    def _get_checked_instance(self, cls, _instance=None):
+        if _instance is None and type(self).__init__ != cls.__init__:
+            raise NotImplementedError(
+                f"Subclass {self.__class__.__name__} of {cls.__name__} that defines a custom __init__ method "
+                "must also define a custom .expand() method."
+            )
+        return self.__new__(type(self)) if _instance is None else _instance
+
+    def __repr__(self) -> str:
+        param_names = [k for k, _ in self.arg_constraints.items() if k in self.__dict__]
+        args_string = ", ".join(
+            [
+                f"{p}: {self.__dict__[p] if self.__dict__[p].numel() == 1 else self.__dict__[p].size()}"
+                for p in param_names
+            ]
+        )
+        return self.__class__.__name__ + "(" + args_string + ")"

rl/Lib/site-packages/torch/distributions/exp_family.py

@@ -0,0 +1,64 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.distributions.distribution import Distribution
+
+
+__all__ = ["ExponentialFamily"]
+
+
+class ExponentialFamily(Distribution):
+    r"""
+    ExponentialFamily is the abstract base class for probability distributions belonging to an
+    exponential family, whose probability mass/density function has the form is defined below
+
+    .. math::
+
+        p_{F}(x; \theta) = \exp(\langle t(x), \theta\rangle - F(\theta) + k(x))
+
+    where :math:`\theta` denotes the natural parameters, :math:`t(x)` denotes the sufficient statistic,
+    :math:`F(\theta)` is the log normalizer function for a given family and :math:`k(x)` is the carrier
+    measure.
+
+    Note:
+        This class is an intermediary between the `Distribution` class and distributions which belong
+        to an exponential family mainly to check the correctness of the `.entropy()` and analytic KL
+        divergence methods. We use this class to compute the entropy and KL divergence using the AD
+        framework and Bregman divergences (courtesy of: Frank Nielsen and Richard Nock, Entropies and
+        Cross-entropies of Exponential Families).
+    """
+
+    @property
+    def _natural_params(self):
+        """
+        Abstract method for natural parameters. Returns a tuple of Tensors based
+        on the distribution
+        """
+        raise NotImplementedError
+
+    def _log_normalizer(self, *natural_params):
+        """
+        Abstract method for log normalizer function. Returns a log normalizer based on
+        the distribution and input
+        """
+        raise NotImplementedError
+
+    @property
+    def _mean_carrier_measure(self):
+        """
+        Abstract method for expected carrier measure, which is required for computing
+        entropy.
+        """
+        raise NotImplementedError
+
+    def entropy(self):
+        """
+        Method to compute the entropy using Bregman divergence of the log normalizer.
+        """
+        result = -self._mean_carrier_measure
+        nparams = [p.detach().requires_grad_() for p in self._natural_params]
+        lg_normal = self._log_normalizer(*nparams)
+        gradients = torch.autograd.grad(lg_normal.sum(), nparams, create_graph=True)
+        result += lg_normal
+        for np, g in zip(nparams, gradients):
+            result -= (np * g).reshape(self._batch_shape + (-1,)).sum(-1)
+        return result

rl/Lib/site-packages/torch/distributions/exponential.py

@@ -0,0 +1,87 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Exponential"]
+
+
+class Exponential(ExponentialFamily):
+    r"""
+    Creates a Exponential distribution parameterized by :attr:`rate`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Exponential(torch.tensor([1.0]))
+        >>> m.sample()  # Exponential distributed with rate=1
+        tensor([ 0.1046])
+
+    Args:
+        rate (float or Tensor): rate = 1 / scale of the distribution
+    """
+    arg_constraints = {"rate": constraints.positive}
+    support = constraints.nonnegative
+    has_rsample = True
+    _mean_carrier_measure = 0
+
+    @property
+    def mean(self):
+        return self.rate.reciprocal()
+
+    @property
+    def mode(self):
+        return torch.zeros_like(self.rate)
+
+    @property
+    def stddev(self):
+        return self.rate.reciprocal()
+
+    @property
+    def variance(self):
+        return self.rate.pow(-2)
+
+    def __init__(self, rate, validate_args=None):
+        (self.rate,) = broadcast_all(rate)
+        batch_shape = torch.Size() if isinstance(rate, Number) else self.rate.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Exponential, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.rate = self.rate.expand(batch_shape)
+        super(Exponential, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        return self.rate.new(shape).exponential_() / self.rate
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return self.rate.log() - self.rate * value
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return 1 - torch.exp(-self.rate * value)
+
+    def icdf(self, value):
+        return -torch.log1p(-value) / self.rate
+
+    def entropy(self):
+        return 1.0 - torch.log(self.rate)
+
+    @property
+    def _natural_params(self):
+        return (-self.rate,)
+
+    def _log_normalizer(self, x):
+        return -torch.log(-x)

rl/Lib/site-packages/torch/distributions/fishersnedecor.py

@@ -0,0 +1,101 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch import nan
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.gamma import Gamma
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["FisherSnedecor"]
+
+
+class FisherSnedecor(Distribution):
+    r"""
+    Creates a Fisher-Snedecor distribution parameterized by :attr:`df1` and :attr:`df2`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = FisherSnedecor(torch.tensor([1.0]), torch.tensor([2.0]))
+        >>> m.sample()  # Fisher-Snedecor-distributed with df1=1 and df2=2
+        tensor([ 0.2453])
+
+    Args:
+        df1 (float or Tensor): degrees of freedom parameter 1
+        df2 (float or Tensor): degrees of freedom parameter 2
+    """
+    arg_constraints = {"df1": constraints.positive, "df2": constraints.positive}
+    support = constraints.positive
+    has_rsample = True
+
+    def __init__(self, df1, df2, validate_args=None):
+        self.df1, self.df2 = broadcast_all(df1, df2)
+        self._gamma1 = Gamma(self.df1 * 0.5, self.df1)
+        self._gamma2 = Gamma(self.df2 * 0.5, self.df2)
+
+        if isinstance(df1, Number) and isinstance(df2, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.df1.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(FisherSnedecor, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.df1 = self.df1.expand(batch_shape)
+        new.df2 = self.df2.expand(batch_shape)
+        new._gamma1 = self._gamma1.expand(batch_shape)
+        new._gamma2 = self._gamma2.expand(batch_shape)
+        super(FisherSnedecor, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def mean(self):
+        df2 = self.df2.clone(memory_format=torch.contiguous_format)
+        df2[df2 <= 2] = nan
+        return df2 / (df2 - 2)
+
+    @property
+    def mode(self):
+        mode = (self.df1 - 2) / self.df1 * self.df2 / (self.df2 + 2)
+        mode[self.df1 <= 2] = nan
+        return mode
+
+    @property
+    def variance(self):
+        df2 = self.df2.clone(memory_format=torch.contiguous_format)
+        df2[df2 <= 4] = nan
+        return (
+            2
+            * df2.pow(2)
+            * (self.df1 + df2 - 2)
+            / (self.df1 * (df2 - 2).pow(2) * (df2 - 4))
+        )
+
+    def rsample(self, sample_shape: _size = torch.Size(())) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        #   X1 ~ Gamma(df1 / 2, 1 / df1), X2 ~ Gamma(df2 / 2, 1 / df2)
+        #   Y = df2 * df1 * X1 / (df1 * df2 * X2) = X1 / X2 ~ F(df1, df2)
+        X1 = self._gamma1.rsample(sample_shape).view(shape)
+        X2 = self._gamma2.rsample(sample_shape).view(shape)
+        tiny = torch.finfo(X2.dtype).tiny
+        X2.clamp_(min=tiny)
+        Y = X1 / X2
+        Y.clamp_(min=tiny)
+        return Y
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        ct1 = self.df1 * 0.5
+        ct2 = self.df2 * 0.5
+        ct3 = self.df1 / self.df2
+        t1 = (ct1 + ct2).lgamma() - ct1.lgamma() - ct2.lgamma()
+        t2 = ct1 * ct3.log() + (ct1 - 1) * torch.log(value)
+        t3 = (ct1 + ct2) * torch.log1p(ct3 * value)
+        return t1 + t2 - t3

rl/Lib/site-packages/torch/distributions/gamma.py

@@ -0,0 +1,111 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Gamma"]
+
+
+def _standard_gamma(concentration):
+    return torch._standard_gamma(concentration)
+
+
+class Gamma(ExponentialFamily):
+    r"""
+    Creates a Gamma distribution parameterized by shape :attr:`concentration` and :attr:`rate`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Gamma(torch.tensor([1.0]), torch.tensor([1.0]))
+        >>> m.sample()  # Gamma distributed with concentration=1 and rate=1
+        tensor([ 0.1046])
+
+    Args:
+        concentration (float or Tensor): shape parameter of the distribution
+            (often referred to as alpha)
+        rate (float or Tensor): rate = 1 / scale of the distribution
+            (often referred to as beta)
+    """
+    arg_constraints = {
+        "concentration": constraints.positive,
+        "rate": constraints.positive,
+    }
+    support = constraints.nonnegative
+    has_rsample = True
+    _mean_carrier_measure = 0
+
+    @property
+    def mean(self):
+        return self.concentration / self.rate
+
+    @property
+    def mode(self):
+        return ((self.concentration - 1) / self.rate).clamp(min=0)
+
+    @property
+    def variance(self):
+        return self.concentration / self.rate.pow(2)
+
+    def __init__(self, concentration, rate, validate_args=None):
+        self.concentration, self.rate = broadcast_all(concentration, rate)
+        if isinstance(concentration, Number) and isinstance(rate, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.concentration.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Gamma, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.concentration = self.concentration.expand(batch_shape)
+        new.rate = self.rate.expand(batch_shape)
+        super(Gamma, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        value = _standard_gamma(self.concentration.expand(shape)) / self.rate.expand(
+            shape
+        )
+        value.detach().clamp_(
+            min=torch.finfo(value.dtype).tiny
+        )  # do not record in autograd graph
+        return value
+
+    def log_prob(self, value):
+        value = torch.as_tensor(value, dtype=self.rate.dtype, device=self.rate.device)
+        if self._validate_args:
+            self._validate_sample(value)
+        return (
+            torch.xlogy(self.concentration, self.rate)
+            + torch.xlogy(self.concentration - 1, value)
+            - self.rate * value
+            - torch.lgamma(self.concentration)
+        )
+
+    def entropy(self):
+        return (
+            self.concentration
+            - torch.log(self.rate)
+            + torch.lgamma(self.concentration)
+            + (1.0 - self.concentration) * torch.digamma(self.concentration)
+        )
+
+    @property
+    def _natural_params(self):
+        return (self.concentration - 1, -self.rate)
+
+    def _log_normalizer(self, x, y):
+        return torch.lgamma(x + 1) + (x + 1) * torch.log(-y.reciprocal())
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return torch.special.gammainc(self.concentration, self.rate * value)

rl/Lib/site-packages/torch/distributions/geometric.py

@@ -0,0 +1,130 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import (
+    broadcast_all,
+    lazy_property,
+    logits_to_probs,
+    probs_to_logits,
+)
+from torch.nn.functional import binary_cross_entropy_with_logits
+
+
+__all__ = ["Geometric"]
+
+
+class Geometric(Distribution):
+    r"""
+    Creates a Geometric distribution parameterized by :attr:`probs`,
+    where :attr:`probs` is the probability of success of Bernoulli trials.
+
+    .. math::
+
+        P(X=k) = (1-p)^{k} p, k = 0, 1, ...
+
+    .. note::
+        :func:`torch.distributions.geometric.Geometric` :math:`(k+1)`-th trial is the first success
+        hence draws samples in :math:`\{0, 1, \ldots\}`, whereas
+        :func:`torch.Tensor.geometric_` `k`-th trial is the first success hence draws samples in :math:`\{1, 2, \ldots\}`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Geometric(torch.tensor([0.3]))
+        >>> m.sample()  # underlying Bernoulli has 30% chance 1; 70% chance 0
+        tensor([ 2.])
+
+    Args:
+        probs (Number, Tensor): the probability of sampling `1`. Must be in range (0, 1]
+        logits (Number, Tensor): the log-odds of sampling `1`.
+    """
+    arg_constraints = {"probs": constraints.unit_interval, "logits": constraints.real}
+    support = constraints.nonnegative_integer
+
+    def __init__(self, probs=None, logits=None, validate_args=None):
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            (self.probs,) = broadcast_all(probs)
+        else:
+            (self.logits,) = broadcast_all(logits)
+        probs_or_logits = probs if probs is not None else logits
+        if isinstance(probs_or_logits, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = probs_or_logits.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+        if self._validate_args and probs is not None:
+            # Add an extra check beyond unit_interval
+            value = self.probs
+            valid = value > 0
+            if not valid.all():
+                invalid_value = value.data[~valid]
+                raise ValueError(
+                    "Expected parameter probs "
+                    f"({type(value).__name__} of shape {tuple(value.shape)}) "
+                    f"of distribution {repr(self)} "
+                    f"to be positive but found invalid values:\n{invalid_value}"
+                )
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Geometric, _instance)
+        batch_shape = torch.Size(batch_shape)
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(batch_shape)
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(batch_shape)
+        super(Geometric, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def mean(self):
+        return 1.0 / self.probs - 1.0
+
+    @property
+    def mode(self):
+        return torch.zeros_like(self.probs)
+
+    @property
+    def variance(self):
+        return (1.0 / self.probs - 1.0) / self.probs
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs, is_binary=True)
+
+    @lazy_property
+    def probs(self):
+        return logits_to_probs(self.logits, is_binary=True)
+
+    def sample(self, sample_shape=torch.Size()):
+        shape = self._extended_shape(sample_shape)
+        tiny = torch.finfo(self.probs.dtype).tiny
+        with torch.no_grad():
+            if torch._C._get_tracing_state():
+                # [JIT WORKAROUND] lack of support for .uniform_()
+                u = torch.rand(shape, dtype=self.probs.dtype, device=self.probs.device)
+                u = u.clamp(min=tiny)
+            else:
+                u = self.probs.new(shape).uniform_(tiny, 1)
+            return (u.log() / (-self.probs).log1p()).floor()
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        value, probs = broadcast_all(value, self.probs)
+        probs = probs.clone(memory_format=torch.contiguous_format)
+        probs[(probs == 1) & (value == 0)] = 0
+        return value * (-probs).log1p() + self.probs.log()
+
+    def entropy(self):
+        return (
+            binary_cross_entropy_with_logits(self.logits, self.probs, reduction="none")
+            / self.probs
+        )

rl/Lib/site-packages/torch/distributions/gumbel.py

@@ -0,0 +1,83 @@
+# mypy: allow-untyped-defs
+import math
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import AffineTransform, ExpTransform
+from torch.distributions.uniform import Uniform
+from torch.distributions.utils import broadcast_all, euler_constant
+
+
+__all__ = ["Gumbel"]
+
+
+class Gumbel(TransformedDistribution):
+    r"""
+    Samples from a Gumbel Distribution.
+
+    Examples::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Gumbel(torch.tensor([1.0]), torch.tensor([2.0]))
+        >>> m.sample()  # sample from Gumbel distribution with loc=1, scale=2
+        tensor([ 1.0124])
+
+    Args:
+        loc (float or Tensor): Location parameter of the distribution
+        scale (float or Tensor): Scale parameter of the distribution
+    """
+    arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
+    support = constraints.real
+
+    def __init__(self, loc, scale, validate_args=None):
+        self.loc, self.scale = broadcast_all(loc, scale)
+        finfo = torch.finfo(self.loc.dtype)
+        if isinstance(loc, Number) and isinstance(scale, Number):
+            base_dist = Uniform(finfo.tiny, 1 - finfo.eps, validate_args=validate_args)
+        else:
+            base_dist = Uniform(
+                torch.full_like(self.loc, finfo.tiny),
+                torch.full_like(self.loc, 1 - finfo.eps),
+                validate_args=validate_args,
+            )
+        transforms = [
+            ExpTransform().inv,
+            AffineTransform(loc=0, scale=-torch.ones_like(self.scale)),
+            ExpTransform().inv,
+            AffineTransform(loc=loc, scale=-self.scale),
+        ]
+        super().__init__(base_dist, transforms, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Gumbel, _instance)
+        new.loc = self.loc.expand(batch_shape)
+        new.scale = self.scale.expand(batch_shape)
+        return super().expand(batch_shape, _instance=new)
+
+    # Explicitly defining the log probability function for Gumbel due to precision issues
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        y = (self.loc - value) / self.scale
+        return (y - y.exp()) - self.scale.log()
+
+    @property
+    def mean(self):
+        return self.loc + self.scale * euler_constant
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @property
+    def stddev(self):
+        return (math.pi / math.sqrt(6)) * self.scale
+
+    @property
+    def variance(self):
+        return self.stddev.pow(2)
+
+    def entropy(self):
+        return self.scale.log() + (1 + euler_constant)

rl/Lib/site-packages/torch/distributions/half_cauchy.py

@@ -0,0 +1,84 @@
+# mypy: allow-untyped-defs
+import math
+
+import torch
+from torch import inf
+from torch.distributions import constraints
+from torch.distributions.cauchy import Cauchy
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import AbsTransform
+
+
+__all__ = ["HalfCauchy"]
+
+
+class HalfCauchy(TransformedDistribution):
+    r"""
+    Creates a half-Cauchy distribution parameterized by `scale` where::
+
+        X ~ Cauchy(0, scale)
+        Y = |X| ~ HalfCauchy(scale)
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = HalfCauchy(torch.tensor([1.0]))
+        >>> m.sample()  # half-cauchy distributed with scale=1
+        tensor([ 2.3214])
+
+    Args:
+        scale (float or Tensor): scale of the full Cauchy distribution
+    """
+    arg_constraints = {"scale": constraints.positive}
+    support = constraints.nonnegative
+    has_rsample = True
+
+    def __init__(self, scale, validate_args=None):
+        base_dist = Cauchy(0, scale, validate_args=False)
+        super().__init__(base_dist, AbsTransform(), validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(HalfCauchy, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def scale(self):
+        return self.base_dist.scale
+
+    @property
+    def mean(self):
+        return torch.full(
+            self._extended_shape(),
+            math.inf,
+            dtype=self.scale.dtype,
+            device=self.scale.device,
+        )
+
+    @property
+    def mode(self):
+        return torch.zeros_like(self.scale)
+
+    @property
+    def variance(self):
+        return self.base_dist.variance
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        value = torch.as_tensor(
+            value, dtype=self.base_dist.scale.dtype, device=self.base_dist.scale.device
+        )
+        log_prob = self.base_dist.log_prob(value) + math.log(2)
+        log_prob = torch.where(value >= 0, log_prob, -inf)
+        return log_prob
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return 2 * self.base_dist.cdf(value) - 1
+
+    def icdf(self, prob):
+        return self.base_dist.icdf((prob + 1) / 2)
+
+    def entropy(self):
+        return self.base_dist.entropy() - math.log(2)

rl/Lib/site-packages/torch/distributions/half_normal.py

@@ -0,0 +1,76 @@
+# mypy: allow-untyped-defs
+import math
+
+import torch
+from torch import inf
+from torch.distributions import constraints
+from torch.distributions.normal import Normal
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import AbsTransform
+
+
+__all__ = ["HalfNormal"]
+
+
+class HalfNormal(TransformedDistribution):
+    r"""
+    Creates a half-normal distribution parameterized by `scale` where::
+
+        X ~ Normal(0, scale)
+        Y = |X| ~ HalfNormal(scale)
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = HalfNormal(torch.tensor([1.0]))
+        >>> m.sample()  # half-normal distributed with scale=1
+        tensor([ 0.1046])
+
+    Args:
+        scale (float or Tensor): scale of the full Normal distribution
+    """
+    arg_constraints = {"scale": constraints.positive}
+    support = constraints.nonnegative
+    has_rsample = True
+
+    def __init__(self, scale, validate_args=None):
+        base_dist = Normal(0, scale, validate_args=False)
+        super().__init__(base_dist, AbsTransform(), validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(HalfNormal, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def scale(self):
+        return self.base_dist.scale
+
+    @property
+    def mean(self):
+        return self.scale * math.sqrt(2 / math.pi)
+
+    @property
+    def mode(self):
+        return torch.zeros_like(self.scale)
+
+    @property
+    def variance(self):
+        return self.scale.pow(2) * (1 - 2 / math.pi)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        log_prob = self.base_dist.log_prob(value) + math.log(2)
+        log_prob = torch.where(value >= 0, log_prob, -inf)
+        return log_prob
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return 2 * self.base_dist.cdf(value) - 1
+
+    def icdf(self, prob):
+        return self.base_dist.icdf((prob + 1) / 2)
+
+    def entropy(self):
+        return self.base_dist.entropy() - math.log(2)

rl/Lib/site-packages/torch/distributions/independent.py

@@ -0,0 +1,128 @@
+# mypy: allow-untyped-defs
+from typing import Dict
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import _sum_rightmost
+from torch.types import _size
+
+
+__all__ = ["Independent"]
+
+
+class Independent(Distribution):
+    r"""
+    Reinterprets some of the batch dims of a distribution as event dims.
+
+    This is mainly useful for changing the shape of the result of
+    :meth:`log_prob`. For example to create a diagonal Normal distribution with
+    the same shape as a Multivariate Normal distribution (so they are
+    interchangeable), you can::
+
+        >>> from torch.distributions.multivariate_normal import MultivariateNormal
+        >>> from torch.distributions.normal import Normal
+        >>> loc = torch.zeros(3)
+        >>> scale = torch.ones(3)
+        >>> mvn = MultivariateNormal(loc, scale_tril=torch.diag(scale))
+        >>> [mvn.batch_shape, mvn.event_shape]
+        [torch.Size([]), torch.Size([3])]
+        >>> normal = Normal(loc, scale)
+        >>> [normal.batch_shape, normal.event_shape]
+        [torch.Size([3]), torch.Size([])]
+        >>> diagn = Independent(normal, 1)
+        >>> [diagn.batch_shape, diagn.event_shape]
+        [torch.Size([]), torch.Size([3])]
+
+    Args:
+        base_distribution (torch.distributions.distribution.Distribution): a
+            base distribution
+        reinterpreted_batch_ndims (int): the number of batch dims to
+            reinterpret as event dims
+    """
+    arg_constraints: Dict[str, constraints.Constraint] = {}
+
+    def __init__(
+        self, base_distribution, reinterpreted_batch_ndims, validate_args=None
+    ):
+        if reinterpreted_batch_ndims > len(base_distribution.batch_shape):
+            raise ValueError(
+                "Expected reinterpreted_batch_ndims <= len(base_distribution.batch_shape), "
+                f"actual {reinterpreted_batch_ndims} vs {len(base_distribution.batch_shape)}"
+            )
+        shape = base_distribution.batch_shape + base_distribution.event_shape
+        event_dim = reinterpreted_batch_ndims + len(base_distribution.event_shape)
+        batch_shape = shape[: len(shape) - event_dim]
+        event_shape = shape[len(shape) - event_dim :]
+        self.base_dist = base_distribution
+        self.reinterpreted_batch_ndims = reinterpreted_batch_ndims
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Independent, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.base_dist = self.base_dist.expand(
+            batch_shape + self.event_shape[: self.reinterpreted_batch_ndims]
+        )
+        new.reinterpreted_batch_ndims = self.reinterpreted_batch_ndims
+        super(Independent, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def has_rsample(self):
+        return self.base_dist.has_rsample
+
+    @property
+    def has_enumerate_support(self):
+        if self.reinterpreted_batch_ndims > 0:
+            return False
+        return self.base_dist.has_enumerate_support
+
+    @constraints.dependent_property
+    def support(self):
+        result = self.base_dist.support
+        if self.reinterpreted_batch_ndims:
+            result = constraints.independent(result, self.reinterpreted_batch_ndims)
+        return result
+
+    @property
+    def mean(self):
+        return self.base_dist.mean
+
+    @property
+    def mode(self):
+        return self.base_dist.mode
+
+    @property
+    def variance(self):
+        return self.base_dist.variance
+
+    def sample(self, sample_shape=torch.Size()):
+        return self.base_dist.sample(sample_shape)
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        return self.base_dist.rsample(sample_shape)
+
+    def log_prob(self, value):
+        log_prob = self.base_dist.log_prob(value)
+        return _sum_rightmost(log_prob, self.reinterpreted_batch_ndims)
+
+    def entropy(self):
+        entropy = self.base_dist.entropy()
+        return _sum_rightmost(entropy, self.reinterpreted_batch_ndims)
+
+    def enumerate_support(self, expand=True):
+        if self.reinterpreted_batch_ndims > 0:
+            raise NotImplementedError(
+                "Enumeration over cartesian product is not implemented"
+            )
+        return self.base_dist.enumerate_support(expand=expand)
+
+    def __repr__(self):
+        return (
+            self.__class__.__name__
+            + f"({self.base_dist}, {self.reinterpreted_batch_ndims})"
+        )

rl/Lib/site-packages/torch/distributions/inverse_gamma.py

@@ -0,0 +1,81 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.distributions import constraints
+from torch.distributions.gamma import Gamma
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import PowerTransform
+
+
+__all__ = ["InverseGamma"]
+
+
+class InverseGamma(TransformedDistribution):
+    r"""
+    Creates an inverse gamma distribution parameterized by :attr:`concentration` and :attr:`rate`
+    where::
+
+        X ~ Gamma(concentration, rate)
+        Y = 1 / X ~ InverseGamma(concentration, rate)
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterinistic")
+        >>> m = InverseGamma(torch.tensor([2.0]), torch.tensor([3.0]))
+        >>> m.sample()
+        tensor([ 1.2953])
+
+    Args:
+        concentration (float or Tensor): shape parameter of the distribution
+            (often referred to as alpha)
+        rate (float or Tensor): rate = 1 / scale of the distribution
+            (often referred to as beta)
+    """
+    arg_constraints = {
+        "concentration": constraints.positive,
+        "rate": constraints.positive,
+    }
+    support = constraints.positive
+    has_rsample = True
+
+    def __init__(self, concentration, rate, validate_args=None):
+        base_dist = Gamma(concentration, rate, validate_args=validate_args)
+        neg_one = -base_dist.rate.new_ones(())
+        super().__init__(
+            base_dist, PowerTransform(neg_one), validate_args=validate_args
+        )
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(InverseGamma, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def concentration(self):
+        return self.base_dist.concentration
+
+    @property
+    def rate(self):
+        return self.base_dist.rate
+
+    @property
+    def mean(self):
+        result = self.rate / (self.concentration - 1)
+        return torch.where(self.concentration > 1, result, torch.inf)
+
+    @property
+    def mode(self):
+        return self.rate / (self.concentration + 1)
+
+    @property
+    def variance(self):
+        result = self.rate.square() / (
+            (self.concentration - 1).square() * (self.concentration - 2)
+        )
+        return torch.where(self.concentration > 2, result, torch.inf)
+
+    def entropy(self):
+        return (
+            self.concentration
+            + self.rate.log()
+            + self.concentration.lgamma()
+            - (1 + self.concentration) * self.concentration.digamma()
+        )

rl/Lib/site-packages/torch/distributions/kl.py

@@ -0,0 +1,972 @@
+# mypy: allow-untyped-defs
+import math
+import warnings
+from functools import total_ordering
+from typing import Callable, Dict, Tuple, Type
+
+import torch
+from torch import inf
+
+from .bernoulli import Bernoulli
+from .beta import Beta
+from .binomial import Binomial
+from .categorical import Categorical
+from .cauchy import Cauchy
+from .continuous_bernoulli import ContinuousBernoulli
+from .dirichlet import Dirichlet
+from .distribution import Distribution
+from .exp_family import ExponentialFamily
+from .exponential import Exponential
+from .gamma import Gamma
+from .geometric import Geometric
+from .gumbel import Gumbel
+from .half_normal import HalfNormal
+from .independent import Independent
+from .laplace import Laplace
+from .lowrank_multivariate_normal import (
+    _batch_lowrank_logdet,
+    _batch_lowrank_mahalanobis,
+    LowRankMultivariateNormal,
+)
+from .multivariate_normal import _batch_mahalanobis, MultivariateNormal
+from .normal import Normal
+from .one_hot_categorical import OneHotCategorical
+from .pareto import Pareto
+from .poisson import Poisson
+from .transformed_distribution import TransformedDistribution
+from .uniform import Uniform
+from .utils import _sum_rightmost, euler_constant as _euler_gamma
+
+
+_KL_REGISTRY: Dict[
+    Tuple[Type, Type], Callable
+] = {}  # Source of truth mapping a few general (type, type) pairs to functions.
+_KL_MEMOIZE: Dict[
+    Tuple[Type, Type], Callable
+] = {}  # Memoized version mapping many specific (type, type) pairs to functions.
+
+__all__ = ["register_kl", "kl_divergence"]
+
+
+def register_kl(type_p, type_q):
+    """
+    Decorator to register a pairwise function with :meth:`kl_divergence`.
+    Usage::
+
+        @register_kl(Normal, Normal)
+        def kl_normal_normal(p, q):
+            # insert implementation here
+
+    Lookup returns the most specific (type,type) match ordered by subclass. If
+    the match is ambiguous, a `RuntimeWarning` is raised. For example to
+    resolve the ambiguous situation::
+
+        @register_kl(BaseP, DerivedQ)
+        def kl_version1(p, q): ...
+        @register_kl(DerivedP, BaseQ)
+        def kl_version2(p, q): ...
+
+    you should register a third most-specific implementation, e.g.::
+
+        register_kl(DerivedP, DerivedQ)(kl_version1)  # Break the tie.
+
+    Args:
+        type_p (type): A subclass of :class:`~torch.distributions.Distribution`.
+        type_q (type): A subclass of :class:`~torch.distributions.Distribution`.
+    """
+    if not isinstance(type_p, type) and issubclass(type_p, Distribution):
+        raise TypeError(
+            f"Expected type_p to be a Distribution subclass but got {type_p}"
+        )
+    if not isinstance(type_q, type) and issubclass(type_q, Distribution):
+        raise TypeError(
+            f"Expected type_q to be a Distribution subclass but got {type_q}"
+        )
+
+    def decorator(fun):
+        _KL_REGISTRY[type_p, type_q] = fun
+        _KL_MEMOIZE.clear()  # reset since lookup order may have changed
+        return fun
+
+    return decorator
+
+
+@total_ordering
+class _Match:
+    __slots__ = ["types"]
+
+    def __init__(self, *types):
+        self.types = types
+
+    def __eq__(self, other):
+        return self.types == other.types
+
+    def __le__(self, other):
+        for x, y in zip(self.types, other.types):
+            if not issubclass(x, y):
+                return False
+            if x is not y:
+                break
+        return True
+
+
+def _dispatch_kl(type_p, type_q):
+    """
+    Find the most specific approximate match, assuming single inheritance.
+    """
+    matches = [
+        (super_p, super_q)
+        for super_p, super_q in _KL_REGISTRY
+        if issubclass(type_p, super_p) and issubclass(type_q, super_q)
+    ]
+    if not matches:
+        return NotImplemented
+    # Check that the left- and right- lexicographic orders agree.
+    # mypy isn't smart enough to know that _Match implements __lt__
+    # see: https://github.com/python/typing/issues/760#issuecomment-710670503
+    left_p, left_q = min(_Match(*m) for m in matches).types  # type: ignore[type-var]
+    right_q, right_p = min(_Match(*reversed(m)) for m in matches).types  # type: ignore[type-var]
+    left_fun = _KL_REGISTRY[left_p, left_q]
+    right_fun = _KL_REGISTRY[right_p, right_q]
+    if left_fun is not right_fun:
+        warnings.warn(
+            f"Ambiguous kl_divergence({type_p.__name__}, {type_q.__name__}). "
+            f"Please register_kl({left_p.__name__}, {right_q.__name__})",
+            RuntimeWarning,
+        )
+    return left_fun
+
+
+def _infinite_like(tensor):
+    """
+    Helper function for obtaining infinite KL Divergence throughout
+    """
+    return torch.full_like(tensor, inf)
+
+
+def _x_log_x(tensor):
+    """
+    Utility function for calculating x log x
+    """
+    return tensor * tensor.log()
+
+
+def _batch_trace_XXT(bmat):
+    """
+    Utility function for calculating the trace of XX^{T} with X having arbitrary trailing batch dimensions
+    """
+    n = bmat.size(-1)
+    m = bmat.size(-2)
+    flat_trace = bmat.reshape(-1, m * n).pow(2).sum(-1)
+    return flat_trace.reshape(bmat.shape[:-2])
+
+
+def kl_divergence(p: Distribution, q: Distribution) -> torch.Tensor:
+    r"""
+    Compute Kullback-Leibler divergence :math:`KL(p \| q)` between two distributions.
+
+    .. math::
+
+        KL(p \| q) = \int p(x) \log\frac {p(x)} {q(x)} \,dx
+
+    Args:
+        p (Distribution): A :class:`~torch.distributions.Distribution` object.
+        q (Distribution): A :class:`~torch.distributions.Distribution` object.
+
+    Returns:
+        Tensor: A batch of KL divergences of shape `batch_shape`.
+
+    Raises:
+        NotImplementedError: If the distribution types have not been registered via
+            :meth:`register_kl`.
+    """
+    try:
+        fun = _KL_MEMOIZE[type(p), type(q)]
+    except KeyError:
+        fun = _dispatch_kl(type(p), type(q))
+        _KL_MEMOIZE[type(p), type(q)] = fun
+    if fun is NotImplemented:
+        raise NotImplementedError(
+            f"No KL(p || q) is implemented for p type {p.__class__.__name__} and q type {q.__class__.__name__}"
+        )
+    return fun(p, q)
+
+
+################################################################################
+# KL Divergence Implementations
+################################################################################
+
+# Same distributions
+
+
+@register_kl(Bernoulli, Bernoulli)
+def _kl_bernoulli_bernoulli(p, q):
+    t1 = p.probs * (
+        torch.nn.functional.softplus(-q.logits)
+        - torch.nn.functional.softplus(-p.logits)
+    )
+    t1[q.probs == 0] = inf
+    t1[p.probs == 0] = 0
+    t2 = (1 - p.probs) * (
+        torch.nn.functional.softplus(q.logits) - torch.nn.functional.softplus(p.logits)
+    )
+    t2[q.probs == 1] = inf
+    t2[p.probs == 1] = 0
+    return t1 + t2
+
+
+@register_kl(Beta, Beta)
+def _kl_beta_beta(p, q):
+    sum_params_p = p.concentration1 + p.concentration0
+    sum_params_q = q.concentration1 + q.concentration0
+    t1 = q.concentration1.lgamma() + q.concentration0.lgamma() + (sum_params_p).lgamma()
+    t2 = p.concentration1.lgamma() + p.concentration0.lgamma() + (sum_params_q).lgamma()
+    t3 = (p.concentration1 - q.concentration1) * torch.digamma(p.concentration1)
+    t4 = (p.concentration0 - q.concentration0) * torch.digamma(p.concentration0)
+    t5 = (sum_params_q - sum_params_p) * torch.digamma(sum_params_p)
+    return t1 - t2 + t3 + t4 + t5
+
+
+@register_kl(Binomial, Binomial)
+def _kl_binomial_binomial(p, q):
+    # from https://math.stackexchange.com/questions/2214993/
+    # kullback-leibler-divergence-for-binomial-distributions-p-and-q
+    if (p.total_count < q.total_count).any():
+        raise NotImplementedError(
+            "KL between Binomials where q.total_count > p.total_count is not implemented"
+        )
+    kl = p.total_count * (
+        p.probs * (p.logits - q.logits) + (-p.probs).log1p() - (-q.probs).log1p()
+    )
+    inf_idxs = p.total_count > q.total_count
+    kl[inf_idxs] = _infinite_like(kl[inf_idxs])
+    return kl
+
+
+@register_kl(Categorical, Categorical)
+def _kl_categorical_categorical(p, q):
+    t = p.probs * (p.logits - q.logits)
+    t[(q.probs == 0).expand_as(t)] = inf
+    t[(p.probs == 0).expand_as(t)] = 0
+    return t.sum(-1)
+
+
+@register_kl(ContinuousBernoulli, ContinuousBernoulli)
+def _kl_continuous_bernoulli_continuous_bernoulli(p, q):
+    t1 = p.mean * (p.logits - q.logits)
+    t2 = p._cont_bern_log_norm() + torch.log1p(-p.probs)
+    t3 = -q._cont_bern_log_norm() - torch.log1p(-q.probs)
+    return t1 + t2 + t3
+
+
+@register_kl(Dirichlet, Dirichlet)
+def _kl_dirichlet_dirichlet(p, q):
+    # From http://bariskurt.com/kullback-leibler-divergence-between-two-dirichlet-and-beta-distributions/
+    sum_p_concentration = p.concentration.sum(-1)
+    sum_q_concentration = q.concentration.sum(-1)
+    t1 = sum_p_concentration.lgamma() - sum_q_concentration.lgamma()
+    t2 = (p.concentration.lgamma() - q.concentration.lgamma()).sum(-1)
+    t3 = p.concentration - q.concentration
+    t4 = p.concentration.digamma() - sum_p_concentration.digamma().unsqueeze(-1)
+    return t1 - t2 + (t3 * t4).sum(-1)
+
+
+@register_kl(Exponential, Exponential)
+def _kl_exponential_exponential(p, q):
+    rate_ratio = q.rate / p.rate
+    t1 = -rate_ratio.log()
+    return t1 + rate_ratio - 1
+
+
+@register_kl(ExponentialFamily, ExponentialFamily)
+def _kl_expfamily_expfamily(p, q):
+    if not type(p) == type(q):
+        raise NotImplementedError(
+            "The cross KL-divergence between different exponential families cannot \
+                            be computed using Bregman divergences"
+        )
+    p_nparams = [np.detach().requires_grad_() for np in p._natural_params]
+    q_nparams = q._natural_params
+    lg_normal = p._log_normalizer(*p_nparams)
+    gradients = torch.autograd.grad(lg_normal.sum(), p_nparams, create_graph=True)
+    result = q._log_normalizer(*q_nparams) - lg_normal
+    for pnp, qnp, g in zip(p_nparams, q_nparams, gradients):
+        term = (qnp - pnp) * g
+        result -= _sum_rightmost(term, len(q.event_shape))
+    return result
+
+
+@register_kl(Gamma, Gamma)
+def _kl_gamma_gamma(p, q):
+    t1 = q.concentration * (p.rate / q.rate).log()
+    t2 = torch.lgamma(q.concentration) - torch.lgamma(p.concentration)
+    t3 = (p.concentration - q.concentration) * torch.digamma(p.concentration)
+    t4 = (q.rate - p.rate) * (p.concentration / p.rate)
+    return t1 + t2 + t3 + t4
+
+
+@register_kl(Gumbel, Gumbel)
+def _kl_gumbel_gumbel(p, q):
+    ct1 = p.scale / q.scale
+    ct2 = q.loc / q.scale
+    ct3 = p.loc / q.scale
+    t1 = -ct1.log() - ct2 + ct3
+    t2 = ct1 * _euler_gamma
+    t3 = torch.exp(ct2 + (1 + ct1).lgamma() - ct3)
+    return t1 + t2 + t3 - (1 + _euler_gamma)
+
+
+@register_kl(Geometric, Geometric)
+def _kl_geometric_geometric(p, q):
+    return -p.entropy() - torch.log1p(-q.probs) / p.probs - q.logits
+
+
+@register_kl(HalfNormal, HalfNormal)
+def _kl_halfnormal_halfnormal(p, q):
+    return _kl_normal_normal(p.base_dist, q.base_dist)
+
+
+@register_kl(Laplace, Laplace)
+def _kl_laplace_laplace(p, q):
+    # From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf
+    scale_ratio = p.scale / q.scale
+    loc_abs_diff = (p.loc - q.loc).abs()
+    t1 = -scale_ratio.log()
+    t2 = loc_abs_diff / q.scale
+    t3 = scale_ratio * torch.exp(-loc_abs_diff / p.scale)
+    return t1 + t2 + t3 - 1
+
+
+@register_kl(LowRankMultivariateNormal, LowRankMultivariateNormal)
+def _kl_lowrankmultivariatenormal_lowrankmultivariatenormal(p, q):
+    if p.event_shape != q.event_shape:
+        raise ValueError(
+            "KL-divergence between two Low Rank Multivariate Normals with\
+                          different event shapes cannot be computed"
+        )
+
+    term1 = _batch_lowrank_logdet(
+        q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag, q._capacitance_tril
+    ) - _batch_lowrank_logdet(
+        p._unbroadcasted_cov_factor, p._unbroadcasted_cov_diag, p._capacitance_tril
+    )
+    term3 = _batch_lowrank_mahalanobis(
+        q._unbroadcasted_cov_factor,
+        q._unbroadcasted_cov_diag,
+        q.loc - p.loc,
+        q._capacitance_tril,
+    )
+    # Expands term2 according to
+    # inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ (pW @ pW.T + pD)
+    #                  = [inv(qD) - A.T @ A] @ (pD + pW @ pW.T)
+    qWt_qDinv = q._unbroadcasted_cov_factor.mT / q._unbroadcasted_cov_diag.unsqueeze(-2)
+    A = torch.linalg.solve_triangular(q._capacitance_tril, qWt_qDinv, upper=False)
+    term21 = (p._unbroadcasted_cov_diag / q._unbroadcasted_cov_diag).sum(-1)
+    term22 = _batch_trace_XXT(
+        p._unbroadcasted_cov_factor * q._unbroadcasted_cov_diag.rsqrt().unsqueeze(-1)
+    )
+    term23 = _batch_trace_XXT(A * p._unbroadcasted_cov_diag.sqrt().unsqueeze(-2))
+    term24 = _batch_trace_XXT(A.matmul(p._unbroadcasted_cov_factor))
+    term2 = term21 + term22 - term23 - term24
+    return 0.5 * (term1 + term2 + term3 - p.event_shape[0])
+
+
+@register_kl(MultivariateNormal, LowRankMultivariateNormal)
+def _kl_multivariatenormal_lowrankmultivariatenormal(p, q):
+    if p.event_shape != q.event_shape:
+        raise ValueError(
+            "KL-divergence between two (Low Rank) Multivariate Normals with\
+                          different event shapes cannot be computed"
+        )
+
+    term1 = _batch_lowrank_logdet(
+        q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag, q._capacitance_tril
+    ) - 2 * p._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)
+    term3 = _batch_lowrank_mahalanobis(
+        q._unbroadcasted_cov_factor,
+        q._unbroadcasted_cov_diag,
+        q.loc - p.loc,
+        q._capacitance_tril,
+    )
+    # Expands term2 according to
+    # inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ p_tril @ p_tril.T
+    #                  = [inv(qD) - A.T @ A] @ p_tril @ p_tril.T
+    qWt_qDinv = q._unbroadcasted_cov_factor.mT / q._unbroadcasted_cov_diag.unsqueeze(-2)
+    A = torch.linalg.solve_triangular(q._capacitance_tril, qWt_qDinv, upper=False)
+    term21 = _batch_trace_XXT(
+        p._unbroadcasted_scale_tril * q._unbroadcasted_cov_diag.rsqrt().unsqueeze(-1)
+    )
+    term22 = _batch_trace_XXT(A.matmul(p._unbroadcasted_scale_tril))
+    term2 = term21 - term22
+    return 0.5 * (term1 + term2 + term3 - p.event_shape[0])
+
+
+@register_kl(LowRankMultivariateNormal, MultivariateNormal)
+def _kl_lowrankmultivariatenormal_multivariatenormal(p, q):
+    if p.event_shape != q.event_shape:
+        raise ValueError(
+            "KL-divergence between two (Low Rank) Multivariate Normals with\
+                          different event shapes cannot be computed"
+        )
+
+    term1 = 2 * q._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(
+        -1
+    ) - _batch_lowrank_logdet(
+        p._unbroadcasted_cov_factor, p._unbroadcasted_cov_diag, p._capacitance_tril
+    )
+    term3 = _batch_mahalanobis(q._unbroadcasted_scale_tril, (q.loc - p.loc))
+    # Expands term2 according to
+    # inv(qcov) @ pcov = inv(q_tril @ q_tril.T) @ (pW @ pW.T + pD)
+    combined_batch_shape = torch._C._infer_size(
+        q._unbroadcasted_scale_tril.shape[:-2], p._unbroadcasted_cov_factor.shape[:-2]
+    )
+    n = p.event_shape[0]
+    q_scale_tril = q._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n))
+    p_cov_factor = p._unbroadcasted_cov_factor.expand(
+        combined_batch_shape + (n, p.cov_factor.size(-1))
+    )
+    p_cov_diag = torch.diag_embed(p._unbroadcasted_cov_diag.sqrt()).expand(
+        combined_batch_shape + (n, n)
+    )
+    term21 = _batch_trace_XXT(
+        torch.linalg.solve_triangular(q_scale_tril, p_cov_factor, upper=False)
+    )
+    term22 = _batch_trace_XXT(
+        torch.linalg.solve_triangular(q_scale_tril, p_cov_diag, upper=False)
+    )
+    term2 = term21 + term22
+    return 0.5 * (term1 + term2 + term3 - p.event_shape[0])
+
+
+@register_kl(MultivariateNormal, MultivariateNormal)
+def _kl_multivariatenormal_multivariatenormal(p, q):
+    # From https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Kullback%E2%80%93Leibler_divergence
+    if p.event_shape != q.event_shape:
+        raise ValueError(
+            "KL-divergence between two Multivariate Normals with\
+                          different event shapes cannot be computed"
+        )
+
+    half_term1 = q._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(
+        -1
+    ) - p._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)
+    combined_batch_shape = torch._C._infer_size(
+        q._unbroadcasted_scale_tril.shape[:-2], p._unbroadcasted_scale_tril.shape[:-2]
+    )
+    n = p.event_shape[0]
+    q_scale_tril = q._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n))
+    p_scale_tril = p._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n))
+    term2 = _batch_trace_XXT(
+        torch.linalg.solve_triangular(q_scale_tril, p_scale_tril, upper=False)
+    )
+    term3 = _batch_mahalanobis(q._unbroadcasted_scale_tril, (q.loc - p.loc))
+    return half_term1 + 0.5 * (term2 + term3 - n)
+
+
+@register_kl(Normal, Normal)
+def _kl_normal_normal(p, q):
+    var_ratio = (p.scale / q.scale).pow(2)
+    t1 = ((p.loc - q.loc) / q.scale).pow(2)
+    return 0.5 * (var_ratio + t1 - 1 - var_ratio.log())
+
+
+@register_kl(OneHotCategorical, OneHotCategorical)
+def _kl_onehotcategorical_onehotcategorical(p, q):
+    return _kl_categorical_categorical(p._categorical, q._categorical)
+
+
+@register_kl(Pareto, Pareto)
+def _kl_pareto_pareto(p, q):
+    # From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf
+    scale_ratio = p.scale / q.scale
+    alpha_ratio = q.alpha / p.alpha
+    t1 = q.alpha * scale_ratio.log()
+    t2 = -alpha_ratio.log()
+    result = t1 + t2 + alpha_ratio - 1
+    result[p.support.lower_bound < q.support.lower_bound] = inf
+    return result
+
+
+@register_kl(Poisson, Poisson)
+def _kl_poisson_poisson(p, q):
+    return p.rate * (p.rate.log() - q.rate.log()) - (p.rate - q.rate)
+
+
+@register_kl(TransformedDistribution, TransformedDistribution)
+def _kl_transformed_transformed(p, q):
+    if p.transforms != q.transforms:
+        raise NotImplementedError
+    if p.event_shape != q.event_shape:
+        raise NotImplementedError
+    return kl_divergence(p.base_dist, q.base_dist)
+
+
+@register_kl(Uniform, Uniform)
+def _kl_uniform_uniform(p, q):
+    result = ((q.high - q.low) / (p.high - p.low)).log()
+    result[(q.low > p.low) | (q.high < p.high)] = inf
+    return result
+
+
+# Different distributions
+@register_kl(Bernoulli, Poisson)
+def _kl_bernoulli_poisson(p, q):
+    return -p.entropy() - (p.probs * q.rate.log() - q.rate)
+
+
+@register_kl(Beta, ContinuousBernoulli)
+def _kl_beta_continuous_bernoulli(p, q):
+    return (
+        -p.entropy()
+        - p.mean * q.logits
+        - torch.log1p(-q.probs)
+        - q._cont_bern_log_norm()
+    )
+
+
+@register_kl(Beta, Pareto)
+def _kl_beta_infinity(p, q):
+    return _infinite_like(p.concentration1)
+
+
+@register_kl(Beta, Exponential)
+def _kl_beta_exponential(p, q):
+    return (
+        -p.entropy()
+        - q.rate.log()
+        + q.rate * (p.concentration1 / (p.concentration1 + p.concentration0))
+    )
+
+
+@register_kl(Beta, Gamma)
+def _kl_beta_gamma(p, q):
+    t1 = -p.entropy()
+    t2 = q.concentration.lgamma() - q.concentration * q.rate.log()
+    t3 = (q.concentration - 1) * (
+        p.concentration1.digamma() - (p.concentration1 + p.concentration0).digamma()
+    )
+    t4 = q.rate * p.concentration1 / (p.concentration1 + p.concentration0)
+    return t1 + t2 - t3 + t4
+
+
+# TODO: Add Beta-Laplace KL Divergence
+
+
+@register_kl(Beta, Normal)
+def _kl_beta_normal(p, q):
+    E_beta = p.concentration1 / (p.concentration1 + p.concentration0)
+    var_normal = q.scale.pow(2)
+    t1 = -p.entropy()
+    t2 = 0.5 * (var_normal * 2 * math.pi).log()
+    t3 = (
+        E_beta * (1 - E_beta) / (p.concentration1 + p.concentration0 + 1)
+        + E_beta.pow(2)
+    ) * 0.5
+    t4 = q.loc * E_beta
+    t5 = q.loc.pow(2) * 0.5
+    return t1 + t2 + (t3 - t4 + t5) / var_normal
+
+
+@register_kl(Beta, Uniform)
+def _kl_beta_uniform(p, q):
+    result = -p.entropy() + (q.high - q.low).log()
+    result[(q.low > p.support.lower_bound) | (q.high < p.support.upper_bound)] = inf
+    return result
+
+
+# Note that the KL between a ContinuousBernoulli and Beta has no closed form
+
+
+@register_kl(ContinuousBernoulli, Pareto)
+def _kl_continuous_bernoulli_infinity(p, q):
+    return _infinite_like(p.probs)
+
+
+@register_kl(ContinuousBernoulli, Exponential)
+def _kl_continuous_bernoulli_exponential(p, q):
+    return -p.entropy() - torch.log(q.rate) + q.rate * p.mean
+
+
+# Note that the KL between a ContinuousBernoulli and Gamma has no closed form
+# TODO: Add ContinuousBernoulli-Laplace KL Divergence
+
+
+@register_kl(ContinuousBernoulli, Normal)
+def _kl_continuous_bernoulli_normal(p, q):
+    t1 = -p.entropy()
+    t2 = 0.5 * (math.log(2.0 * math.pi) + torch.square(q.loc / q.scale)) + torch.log(
+        q.scale
+    )
+    t3 = (p.variance + torch.square(p.mean) - 2.0 * q.loc * p.mean) / (
+        2.0 * torch.square(q.scale)
+    )
+    return t1 + t2 + t3
+
+
+@register_kl(ContinuousBernoulli, Uniform)
+def _kl_continuous_bernoulli_uniform(p, q):
+    result = -p.entropy() + (q.high - q.low).log()
+    return torch.where(
+        torch.max(
+            torch.ge(q.low, p.support.lower_bound),
+            torch.le(q.high, p.support.upper_bound),
+        ),
+        torch.ones_like(result) * inf,
+        result,
+    )
+
+
+@register_kl(Exponential, Beta)
+@register_kl(Exponential, ContinuousBernoulli)
+@register_kl(Exponential, Pareto)
+@register_kl(Exponential, Uniform)
+def _kl_exponential_infinity(p, q):
+    return _infinite_like(p.rate)
+
+
+@register_kl(Exponential, Gamma)
+def _kl_exponential_gamma(p, q):
+    ratio = q.rate / p.rate
+    t1 = -q.concentration * torch.log(ratio)
+    return (
+        t1
+        + ratio
+        + q.concentration.lgamma()
+        + q.concentration * _euler_gamma
+        - (1 + _euler_gamma)
+    )
+
+
+@register_kl(Exponential, Gumbel)
+def _kl_exponential_gumbel(p, q):
+    scale_rate_prod = p.rate * q.scale
+    loc_scale_ratio = q.loc / q.scale
+    t1 = scale_rate_prod.log() - 1
+    t2 = torch.exp(loc_scale_ratio) * scale_rate_prod / (scale_rate_prod + 1)
+    t3 = scale_rate_prod.reciprocal()
+    return t1 - loc_scale_ratio + t2 + t3
+
+
+# TODO: Add Exponential-Laplace KL Divergence
+
+
+@register_kl(Exponential, Normal)
+def _kl_exponential_normal(p, q):
+    var_normal = q.scale.pow(2)
+    rate_sqr = p.rate.pow(2)
+    t1 = 0.5 * torch.log(rate_sqr * var_normal * 2 * math.pi)
+    t2 = rate_sqr.reciprocal()
+    t3 = q.loc / p.rate
+    t4 = q.loc.pow(2) * 0.5
+    return t1 - 1 + (t2 - t3 + t4) / var_normal
+
+
+@register_kl(Gamma, Beta)
+@register_kl(Gamma, ContinuousBernoulli)
+@register_kl(Gamma, Pareto)
+@register_kl(Gamma, Uniform)
+def _kl_gamma_infinity(p, q):
+    return _infinite_like(p.concentration)
+
+
+@register_kl(Gamma, Exponential)
+def _kl_gamma_exponential(p, q):
+    return -p.entropy() - q.rate.log() + q.rate * p.concentration / p.rate
+
+
+@register_kl(Gamma, Gumbel)
+def _kl_gamma_gumbel(p, q):
+    beta_scale_prod = p.rate * q.scale
+    loc_scale_ratio = q.loc / q.scale
+    t1 = (
+        (p.concentration - 1) * p.concentration.digamma()
+        - p.concentration.lgamma()
+        - p.concentration
+    )
+    t2 = beta_scale_prod.log() + p.concentration / beta_scale_prod
+    t3 = (
+        torch.exp(loc_scale_ratio)
+        * (1 + beta_scale_prod.reciprocal()).pow(-p.concentration)
+        - loc_scale_ratio
+    )
+    return t1 + t2 + t3
+
+
+# TODO: Add Gamma-Laplace KL Divergence
+
+
+@register_kl(Gamma, Normal)
+def _kl_gamma_normal(p, q):
+    var_normal = q.scale.pow(2)
+    beta_sqr = p.rate.pow(2)
+    t1 = (
+        0.5 * torch.log(beta_sqr * var_normal * 2 * math.pi)
+        - p.concentration
+        - p.concentration.lgamma()
+    )
+    t2 = 0.5 * (p.concentration.pow(2) + p.concentration) / beta_sqr
+    t3 = q.loc * p.concentration / p.rate
+    t4 = 0.5 * q.loc.pow(2)
+    return (
+        t1
+        + (p.concentration - 1) * p.concentration.digamma()
+        + (t2 - t3 + t4) / var_normal
+    )
+
+
+@register_kl(Gumbel, Beta)
+@register_kl(Gumbel, ContinuousBernoulli)
+@register_kl(Gumbel, Exponential)
+@register_kl(Gumbel, Gamma)
+@register_kl(Gumbel, Pareto)
+@register_kl(Gumbel, Uniform)
+def _kl_gumbel_infinity(p, q):
+    return _infinite_like(p.loc)
+
+
+# TODO: Add Gumbel-Laplace KL Divergence
+
+
+@register_kl(Gumbel, Normal)
+def _kl_gumbel_normal(p, q):
+    param_ratio = p.scale / q.scale
+    t1 = (param_ratio / math.sqrt(2 * math.pi)).log()
+    t2 = (math.pi * param_ratio * 0.5).pow(2) / 3
+    t3 = ((p.loc + p.scale * _euler_gamma - q.loc) / q.scale).pow(2) * 0.5
+    return -t1 + t2 + t3 - (_euler_gamma + 1)
+
+
+@register_kl(Laplace, Beta)
+@register_kl(Laplace, ContinuousBernoulli)
+@register_kl(Laplace, Exponential)
+@register_kl(Laplace, Gamma)
+@register_kl(Laplace, Pareto)
+@register_kl(Laplace, Uniform)
+def _kl_laplace_infinity(p, q):
+    return _infinite_like(p.loc)
+
+
+@register_kl(Laplace, Normal)
+def _kl_laplace_normal(p, q):
+    var_normal = q.scale.pow(2)
+    scale_sqr_var_ratio = p.scale.pow(2) / var_normal
+    t1 = 0.5 * torch.log(2 * scale_sqr_var_ratio / math.pi)
+    t2 = 0.5 * p.loc.pow(2)
+    t3 = p.loc * q.loc
+    t4 = 0.5 * q.loc.pow(2)
+    return -t1 + scale_sqr_var_ratio + (t2 - t3 + t4) / var_normal - 1
+
+
+@register_kl(Normal, Beta)
+@register_kl(Normal, ContinuousBernoulli)
+@register_kl(Normal, Exponential)
+@register_kl(Normal, Gamma)
+@register_kl(Normal, Pareto)
+@register_kl(Normal, Uniform)
+def _kl_normal_infinity(p, q):
+    return _infinite_like(p.loc)
+
+
+@register_kl(Normal, Gumbel)
+def _kl_normal_gumbel(p, q):
+    mean_scale_ratio = p.loc / q.scale
+    var_scale_sqr_ratio = (p.scale / q.scale).pow(2)
+    loc_scale_ratio = q.loc / q.scale
+    t1 = var_scale_sqr_ratio.log() * 0.5
+    t2 = mean_scale_ratio - loc_scale_ratio
+    t3 = torch.exp(-mean_scale_ratio + 0.5 * var_scale_sqr_ratio + loc_scale_ratio)
+    return -t1 + t2 + t3 - (0.5 * (1 + math.log(2 * math.pi)))
+
+
+@register_kl(Normal, Laplace)
+def _kl_normal_laplace(p, q):
+    loc_diff = p.loc - q.loc
+    scale_ratio = p.scale / q.scale
+    loc_diff_scale_ratio = loc_diff / p.scale
+    t1 = torch.log(scale_ratio)
+    t2 = (
+        math.sqrt(2 / math.pi) * p.scale * torch.exp(-0.5 * loc_diff_scale_ratio.pow(2))
+    )
+    t3 = loc_diff * torch.erf(math.sqrt(0.5) * loc_diff_scale_ratio)
+    return -t1 + (t2 + t3) / q.scale - (0.5 * (1 + math.log(0.5 * math.pi)))
+
+
+@register_kl(Pareto, Beta)
+@register_kl(Pareto, ContinuousBernoulli)
+@register_kl(Pareto, Uniform)
+def _kl_pareto_infinity(p, q):
+    return _infinite_like(p.scale)
+
+
+@register_kl(Pareto, Exponential)
+def _kl_pareto_exponential(p, q):
+    scale_rate_prod = p.scale * q.rate
+    t1 = (p.alpha / scale_rate_prod).log()
+    t2 = p.alpha.reciprocal()
+    t3 = p.alpha * scale_rate_prod / (p.alpha - 1)
+    result = t1 - t2 + t3 - 1
+    result[p.alpha <= 1] = inf
+    return result
+
+
+@register_kl(Pareto, Gamma)
+def _kl_pareto_gamma(p, q):
+    common_term = p.scale.log() + p.alpha.reciprocal()
+    t1 = p.alpha.log() - common_term
+    t2 = q.concentration.lgamma() - q.concentration * q.rate.log()
+    t3 = (1 - q.concentration) * common_term
+    t4 = q.rate * p.alpha * p.scale / (p.alpha - 1)
+    result = t1 + t2 + t3 + t4 - 1
+    result[p.alpha <= 1] = inf
+    return result
+
+
+# TODO: Add Pareto-Laplace KL Divergence
+
+
+@register_kl(Pareto, Normal)
+def _kl_pareto_normal(p, q):
+    var_normal = 2 * q.scale.pow(2)
+    common_term = p.scale / (p.alpha - 1)
+    t1 = (math.sqrt(2 * math.pi) * q.scale * p.alpha / p.scale).log()
+    t2 = p.alpha.reciprocal()
+    t3 = p.alpha * common_term.pow(2) / (p.alpha - 2)
+    t4 = (p.alpha * common_term - q.loc).pow(2)
+    result = t1 - t2 + (t3 + t4) / var_normal - 1
+    result[p.alpha <= 2] = inf
+    return result
+
+
+@register_kl(Poisson, Bernoulli)
+@register_kl(Poisson, Binomial)
+def _kl_poisson_infinity(p, q):
+    return _infinite_like(p.rate)
+
+
+@register_kl(Uniform, Beta)
+def _kl_uniform_beta(p, q):
+    common_term = p.high - p.low
+    t1 = torch.log(common_term)
+    t2 = (
+        (q.concentration1 - 1)
+        * (_x_log_x(p.high) - _x_log_x(p.low) - common_term)
+        / common_term
+    )
+    t3 = (
+        (q.concentration0 - 1)
+        * (_x_log_x(1 - p.high) - _x_log_x(1 - p.low) + common_term)
+        / common_term
+    )
+    t4 = (
+        q.concentration1.lgamma()
+        + q.concentration0.lgamma()
+        - (q.concentration1 + q.concentration0).lgamma()
+    )
+    result = t3 + t4 - t1 - t2
+    result[(p.high > q.support.upper_bound) | (p.low < q.support.lower_bound)] = inf
+    return result
+
+
+@register_kl(Uniform, ContinuousBernoulli)
+def _kl_uniform_continuous_bernoulli(p, q):
+    result = (
+        -p.entropy()
+        - p.mean * q.logits
+        - torch.log1p(-q.probs)
+        - q._cont_bern_log_norm()
+    )
+    return torch.where(
+        torch.max(
+            torch.ge(p.high, q.support.upper_bound),
+            torch.le(p.low, q.support.lower_bound),
+        ),
+        torch.ones_like(result) * inf,
+        result,
+    )
+
+
+@register_kl(Uniform, Exponential)
+def _kl_uniform_exponetial(p, q):
+    result = q.rate * (p.high + p.low) / 2 - ((p.high - p.low) * q.rate).log()
+    result[p.low < q.support.lower_bound] = inf
+    return result
+
+
+@register_kl(Uniform, Gamma)
+def _kl_uniform_gamma(p, q):
+    common_term = p.high - p.low
+    t1 = common_term.log()
+    t2 = q.concentration.lgamma() - q.concentration * q.rate.log()
+    t3 = (
+        (1 - q.concentration)
+        * (_x_log_x(p.high) - _x_log_x(p.low) - common_term)
+        / common_term
+    )
+    t4 = q.rate * (p.high + p.low) / 2
+    result = -t1 + t2 + t3 + t4
+    result[p.low < q.support.lower_bound] = inf
+    return result
+
+
+@register_kl(Uniform, Gumbel)
+def _kl_uniform_gumbel(p, q):
+    common_term = q.scale / (p.high - p.low)
+    high_loc_diff = (p.high - q.loc) / q.scale
+    low_loc_diff = (p.low - q.loc) / q.scale
+    t1 = common_term.log() + 0.5 * (high_loc_diff + low_loc_diff)
+    t2 = common_term * (torch.exp(-high_loc_diff) - torch.exp(-low_loc_diff))
+    return t1 - t2
+
+
+# TODO: Uniform-Laplace KL Divergence
+
+
+@register_kl(Uniform, Normal)
+def _kl_uniform_normal(p, q):
+    common_term = p.high - p.low
+    t1 = (math.sqrt(math.pi * 2) * q.scale / common_term).log()
+    t2 = (common_term).pow(2) / 12
+    t3 = ((p.high + p.low - 2 * q.loc) / 2).pow(2)
+    return t1 + 0.5 * (t2 + t3) / q.scale.pow(2)
+
+
+@register_kl(Uniform, Pareto)
+def _kl_uniform_pareto(p, q):
+    support_uniform = p.high - p.low
+    t1 = (q.alpha * q.scale.pow(q.alpha) * (support_uniform)).log()
+    t2 = (_x_log_x(p.high) - _x_log_x(p.low) - support_uniform) / support_uniform
+    result = t2 * (q.alpha + 1) - t1
+    result[p.low < q.support.lower_bound] = inf
+    return result
+
+
+@register_kl(Independent, Independent)
+def _kl_independent_independent(p, q):
+    if p.reinterpreted_batch_ndims != q.reinterpreted_batch_ndims:
+        raise NotImplementedError
+    result = kl_divergence(p.base_dist, q.base_dist)
+    return _sum_rightmost(result, p.reinterpreted_batch_ndims)
+
+
+@register_kl(Cauchy, Cauchy)
+def _kl_cauchy_cauchy(p, q):
+    # From https://arxiv.org/abs/1905.10965
+    t1 = ((p.scale + q.scale).pow(2) + (p.loc - q.loc).pow(2)).log()
+    t2 = (4 * p.scale * q.scale).log()
+    return t1 - t2
+
+
+def _add_kl_info():
+    """Appends a list of implemented KL functions to the doc for kl_divergence."""
+    rows = [
+        "KL divergence is currently implemented for the following distribution pairs:"
+    ]
+    for p, q in sorted(
+        _KL_REGISTRY, key=lambda p_q: (p_q[0].__name__, p_q[1].__name__)
+    ):
+        rows.append(
+            f"* :class:`~torch.distributions.{p.__name__}` and :class:`~torch.distributions.{q.__name__}`"
+        )
+    kl_info = "\n\t".join(rows)
+    if kl_divergence.__doc__:
+        kl_divergence.__doc__ += kl_info  # type: ignore[operator]

rl/Lib/site-packages/torch/distributions/kumaraswamy.py

@@ -0,0 +1,99 @@
+# mypy: allow-untyped-defs
+import torch
+from torch import nan
+from torch.distributions import constraints
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import AffineTransform, PowerTransform
+from torch.distributions.uniform import Uniform
+from torch.distributions.utils import broadcast_all, euler_constant
+
+
+__all__ = ["Kumaraswamy"]
+
+
+def _moments(a, b, n):
+    """
+    Computes nth moment of Kumaraswamy using using torch.lgamma
+    """
+    arg1 = 1 + n / a
+    log_value = torch.lgamma(arg1) + torch.lgamma(b) - torch.lgamma(arg1 + b)
+    return b * torch.exp(log_value)
+
+
+class Kumaraswamy(TransformedDistribution):
+    r"""
+    Samples from a Kumaraswamy distribution.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Kumaraswamy(torch.tensor([1.0]), torch.tensor([1.0]))
+        >>> m.sample()  # sample from a Kumaraswamy distribution with concentration alpha=1 and beta=1
+        tensor([ 0.1729])
+
+    Args:
+        concentration1 (float or Tensor): 1st concentration parameter of the distribution
+            (often referred to as alpha)
+        concentration0 (float or Tensor): 2nd concentration parameter of the distribution
+            (often referred to as beta)
+    """
+    arg_constraints = {
+        "concentration1": constraints.positive,
+        "concentration0": constraints.positive,
+    }
+    support = constraints.unit_interval
+    has_rsample = True
+
+    def __init__(self, concentration1, concentration0, validate_args=None):
+        self.concentration1, self.concentration0 = broadcast_all(
+            concentration1, concentration0
+        )
+        finfo = torch.finfo(self.concentration0.dtype)
+        base_dist = Uniform(
+            torch.full_like(self.concentration0, 0),
+            torch.full_like(self.concentration0, 1),
+            validate_args=validate_args,
+        )
+        transforms = [
+            PowerTransform(exponent=self.concentration0.reciprocal()),
+            AffineTransform(loc=1.0, scale=-1.0),
+            PowerTransform(exponent=self.concentration1.reciprocal()),
+        ]
+        super().__init__(base_dist, transforms, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Kumaraswamy, _instance)
+        new.concentration1 = self.concentration1.expand(batch_shape)
+        new.concentration0 = self.concentration0.expand(batch_shape)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def mean(self):
+        return _moments(self.concentration1, self.concentration0, 1)
+
+    @property
+    def mode(self):
+        # Evaluate in log-space for numerical stability.
+        log_mode = (
+            self.concentration0.reciprocal() * (-self.concentration0).log1p()
+            - (-self.concentration0 * self.concentration1).log1p()
+        )
+        log_mode[(self.concentration0 < 1) | (self.concentration1 < 1)] = nan
+        return log_mode.exp()
+
+    @property
+    def variance(self):
+        return _moments(self.concentration1, self.concentration0, 2) - torch.pow(
+            self.mean, 2
+        )
+
+    def entropy(self):
+        t1 = 1 - self.concentration1.reciprocal()
+        t0 = 1 - self.concentration0.reciprocal()
+        H0 = torch.digamma(self.concentration0 + 1) + euler_constant
+        return (
+            t0
+            + t1 * H0
+            - torch.log(self.concentration1)
+            - torch.log(self.concentration0)
+        )

rl/Lib/site-packages/torch/distributions/laplace.py

@@ -0,0 +1,97 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Laplace"]
+
+
+class Laplace(Distribution):
+    r"""
+    Creates a Laplace distribution parameterized by :attr:`loc` and :attr:`scale`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Laplace(torch.tensor([0.0]), torch.tensor([1.0]))
+        >>> m.sample()  # Laplace distributed with loc=0, scale=1
+        tensor([ 0.1046])
+
+    Args:
+        loc (float or Tensor): mean of the distribution
+        scale (float or Tensor): scale of the distribution
+    """
+    arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
+    support = constraints.real
+    has_rsample = True
+
+    @property
+    def mean(self):
+        return self.loc
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @property
+    def variance(self):
+        return 2 * self.scale.pow(2)
+
+    @property
+    def stddev(self):
+        return (2**0.5) * self.scale
+
+    def __init__(self, loc, scale, validate_args=None):
+        self.loc, self.scale = broadcast_all(loc, scale)
+        if isinstance(loc, Number) and isinstance(scale, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.loc.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Laplace, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.loc = self.loc.expand(batch_shape)
+        new.scale = self.scale.expand(batch_shape)
+        super(Laplace, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        finfo = torch.finfo(self.loc.dtype)
+        if torch._C._get_tracing_state():
+            # [JIT WORKAROUND] lack of support for .uniform_()
+            u = torch.rand(shape, dtype=self.loc.dtype, device=self.loc.device) * 2 - 1
+            return self.loc - self.scale * u.sign() * torch.log1p(
+                -u.abs().clamp(min=finfo.tiny)
+            )
+        u = self.loc.new(shape).uniform_(finfo.eps - 1, 1)
+        # TODO: If we ever implement tensor.nextafter, below is what we want ideally.
+        # u = self.loc.new(shape).uniform_(self.loc.nextafter(-.5, 0), .5)
+        return self.loc - self.scale * u.sign() * torch.log1p(-u.abs())
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return -torch.log(2 * self.scale) - torch.abs(value - self.loc) / self.scale
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return 0.5 - 0.5 * (value - self.loc).sign() * torch.expm1(
+            -(value - self.loc).abs() / self.scale
+        )
+
+    def icdf(self, value):
+        term = value - 0.5
+        return self.loc - self.scale * (term).sign() * torch.log1p(-2 * term.abs())
+
+    def entropy(self):
+        return 1 + torch.log(2 * self.scale)

rl/Lib/site-packages/torch/distributions/lkj_cholesky.py

@@ -0,0 +1,144 @@
+# mypy: allow-untyped-defs
+"""
+This closely follows the implementation in NumPyro (https://github.com/pyro-ppl/numpyro).
+
+Original copyright notice:
+
+# Copyright: Contributors to the Pyro project.
+# SPDX-License-Identifier: Apache-2.0
+"""
+
+import math
+
+import torch
+from torch.distributions import Beta, constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import broadcast_all
+
+
+__all__ = ["LKJCholesky"]
+
+
+class LKJCholesky(Distribution):
+    r"""
+    LKJ distribution for lower Cholesky factor of correlation matrices.
+    The distribution is controlled by ``concentration`` parameter :math:`\eta`
+    to make the probability of the correlation matrix :math:`M` generated from
+    a Cholesky factor proportional to :math:`\det(M)^{\eta - 1}`. Because of that,
+    when ``concentration == 1``, we have a uniform distribution over Cholesky
+    factors of correlation matrices::
+
+        L ~ LKJCholesky(dim, concentration)
+        X = L @ L' ~ LKJCorr(dim, concentration)
+
+    Note that this distribution samples the
+    Cholesky factor of correlation matrices and not the correlation matrices
+    themselves and thereby differs slightly from the derivations in [1] for
+    the `LKJCorr` distribution. For sampling, this uses the Onion method from
+    [1] Section 3.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> l = LKJCholesky(3, 0.5)
+        >>> l.sample()  # l @ l.T is a sample of a correlation 3x3 matrix
+        tensor([[ 1.0000,  0.0000,  0.0000],
+                [ 0.3516,  0.9361,  0.0000],
+                [-0.1899,  0.4748,  0.8593]])
+
+    Args:
+        dimension (dim): dimension of the matrices
+        concentration (float or Tensor): concentration/shape parameter of the
+            distribution (often referred to as eta)
+
+    **References**
+
+    [1] `Generating random correlation matrices based on vines and extended onion method` (2009),
+    Daniel Lewandowski, Dorota Kurowicka, Harry Joe.
+    Journal of Multivariate Analysis. 100. 10.1016/j.jmva.2009.04.008
+    """
+    arg_constraints = {"concentration": constraints.positive}
+    support = constraints.corr_cholesky
+
+    def __init__(self, dim, concentration=1.0, validate_args=None):
+        if dim < 2:
+            raise ValueError(
+                f"Expected dim to be an integer greater than or equal to 2. Found dim={dim}."
+            )
+        self.dim = dim
+        (self.concentration,) = broadcast_all(concentration)
+        batch_shape = self.concentration.size()
+        event_shape = torch.Size((dim, dim))
+        # This is used to draw vectorized samples from the beta distribution in Sec. 3.2 of [1].
+        marginal_conc = self.concentration + 0.5 * (self.dim - 2)
+        offset = torch.arange(
+            self.dim - 1,
+            dtype=self.concentration.dtype,
+            device=self.concentration.device,
+        )
+        offset = torch.cat([offset.new_zeros((1,)), offset])
+        beta_conc1 = offset + 0.5
+        beta_conc0 = marginal_conc.unsqueeze(-1) - 0.5 * offset
+        self._beta = Beta(beta_conc1, beta_conc0)
+        super().__init__(batch_shape, event_shape, validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(LKJCholesky, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.dim = self.dim
+        new.concentration = self.concentration.expand(batch_shape)
+        new._beta = self._beta.expand(batch_shape + (self.dim,))
+        super(LKJCholesky, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    def sample(self, sample_shape=torch.Size()):
+        # This uses the Onion method, but there are a few differences from [1] Sec. 3.2:
+        # - This vectorizes the for loop and also works for heterogeneous eta.
+        # - Same algorithm generalizes to n=1.
+        # - The procedure is simplified since we are sampling the cholesky factor of
+        #   the correlation matrix instead of the correlation matrix itself. As such,
+        #   we only need to generate `w`.
+        y = self._beta.sample(sample_shape).unsqueeze(-1)
+        u_normal = torch.randn(
+            self._extended_shape(sample_shape), dtype=y.dtype, device=y.device
+        ).tril(-1)
+        u_hypersphere = u_normal / u_normal.norm(dim=-1, keepdim=True)
+        # Replace NaNs in first row
+        u_hypersphere[..., 0, :].fill_(0.0)
+        w = torch.sqrt(y) * u_hypersphere
+        # Fill diagonal elements; clamp for numerical stability
+        eps = torch.finfo(w.dtype).tiny
+        diag_elems = torch.clamp(1 - torch.sum(w**2, dim=-1), min=eps).sqrt()
+        w += torch.diag_embed(diag_elems)
+        return w
+
+    def log_prob(self, value):
+        # See: https://mc-stan.org/docs/2_25/functions-reference/cholesky-lkj-correlation-distribution.html
+        # The probability of a correlation matrix is proportional to
+        #   determinant ** (concentration - 1) = prod(L_ii ^ 2(concentration - 1))
+        # Additionally, the Jacobian of the transformation from Cholesky factor to
+        # correlation matrix is:
+        #   prod(L_ii ^ (D - i))
+        # So the probability of a Cholesky factor is propotional to
+        #   prod(L_ii ^ (2 * concentration - 2 + D - i)) = prod(L_ii ^ order_i)
+        # with order_i = 2 * concentration - 2 + D - i
+        if self._validate_args:
+            self._validate_sample(value)
+        diag_elems = value.diagonal(dim1=-1, dim2=-2)[..., 1:]
+        order = torch.arange(2, self.dim + 1, device=self.concentration.device)
+        order = 2 * (self.concentration - 1).unsqueeze(-1) + self.dim - order
+        unnormalized_log_pdf = torch.sum(order * diag_elems.log(), dim=-1)
+        # Compute normalization constant (page 1999 of [1])
+        dm1 = self.dim - 1
+        alpha = self.concentration + 0.5 * dm1
+        denominator = torch.lgamma(alpha) * dm1
+        numerator = torch.mvlgamma(alpha - 0.5, dm1)
+        # pi_constant in [1] is D * (D - 1) / 4 * log(pi)
+        # pi_constant in multigammaln is (D - 1) * (D - 2) / 4 * log(pi)
+        # hence, we need to add a pi_constant = (D - 1) * log(pi) / 2
+        pi_constant = 0.5 * dm1 * math.log(math.pi)
+        normalize_term = pi_constant + numerator - denominator
+        return unnormalized_log_pdf - normalize_term

rl/Lib/site-packages/torch/distributions/log_normal.py

@@ -0,0 +1,64 @@
+# mypy: allow-untyped-defs
+from torch.distributions import constraints
+from torch.distributions.normal import Normal
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import ExpTransform
+
+
+__all__ = ["LogNormal"]
+
+
+class LogNormal(TransformedDistribution):
+    r"""
+    Creates a log-normal distribution parameterized by
+    :attr:`loc` and :attr:`scale` where::
+
+        X ~ Normal(loc, scale)
+        Y = exp(X) ~ LogNormal(loc, scale)
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = LogNormal(torch.tensor([0.0]), torch.tensor([1.0]))
+        >>> m.sample()  # log-normal distributed with mean=0 and stddev=1
+        tensor([ 0.1046])
+
+    Args:
+        loc (float or Tensor): mean of log of distribution
+        scale (float or Tensor): standard deviation of log of the distribution
+    """
+    arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
+    support = constraints.positive
+    has_rsample = True
+
+    def __init__(self, loc, scale, validate_args=None):
+        base_dist = Normal(loc, scale, validate_args=validate_args)
+        super().__init__(base_dist, ExpTransform(), validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(LogNormal, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def loc(self):
+        return self.base_dist.loc
+
+    @property
+    def scale(self):
+        return self.base_dist.scale
+
+    @property
+    def mean(self):
+        return (self.loc + self.scale.pow(2) / 2).exp()
+
+    @property
+    def mode(self):
+        return (self.loc - self.scale.square()).exp()
+
+    @property
+    def variance(self):
+        scale_sq = self.scale.pow(2)
+        return scale_sq.expm1() * (2 * self.loc + scale_sq).exp()
+
+    def entropy(self):
+        return self.base_dist.entropy() + self.loc

rl/Lib/site-packages/torch/distributions/logistic_normal.py

@@ -0,0 +1,56 @@
+# mypy: allow-untyped-defs
+from torch.distributions import constraints
+from torch.distributions.normal import Normal
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import StickBreakingTransform
+
+
+__all__ = ["LogisticNormal"]
+
+
+class LogisticNormal(TransformedDistribution):
+    r"""
+    Creates a logistic-normal distribution parameterized by :attr:`loc` and :attr:`scale`
+    that define the base `Normal` distribution transformed with the
+    `StickBreakingTransform` such that::
+
+        X ~ LogisticNormal(loc, scale)
+        Y = log(X / (1 - X.cumsum(-1)))[..., :-1] ~ Normal(loc, scale)
+
+    Args:
+        loc (float or Tensor): mean of the base distribution
+        scale (float or Tensor): standard deviation of the base distribution
+
+    Example::
+
+        >>> # logistic-normal distributed with mean=(0, 0, 0) and stddev=(1, 1, 1)
+        >>> # of the base Normal distribution
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = LogisticNormal(torch.tensor([0.0] * 3), torch.tensor([1.0] * 3))
+        >>> m.sample()
+        tensor([ 0.7653,  0.0341,  0.0579,  0.1427])
+
+    """
+    arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
+    support = constraints.simplex
+    has_rsample = True
+
+    def __init__(self, loc, scale, validate_args=None):
+        base_dist = Normal(loc, scale, validate_args=validate_args)
+        if not base_dist.batch_shape:
+            base_dist = base_dist.expand([1])
+        super().__init__(
+            base_dist, StickBreakingTransform(), validate_args=validate_args
+        )
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(LogisticNormal, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def loc(self):
+        return self.base_dist.base_dist.loc
+
+    @property
+    def scale(self):
+        return self.base_dist.base_dist.scale

rl/Lib/site-packages/torch/distributions/lowrank_multivariate_normal.py

@@ -0,0 +1,240 @@
+# mypy: allow-untyped-defs
+import math
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.multivariate_normal import _batch_mahalanobis, _batch_mv
+from torch.distributions.utils import _standard_normal, lazy_property
+from torch.types import _size
+
+
+__all__ = ["LowRankMultivariateNormal"]
+
+
+def _batch_capacitance_tril(W, D):
+    r"""
+    Computes Cholesky of :math:`I + W.T @ inv(D) @ W` for a batch of matrices :math:`W`
+    and a batch of vectors :math:`D`.
+    """
+    m = W.size(-1)
+    Wt_Dinv = W.mT / D.unsqueeze(-2)
+    K = torch.matmul(Wt_Dinv, W).contiguous()
+    K.view(-1, m * m)[:, :: m + 1] += 1  # add identity matrix to K
+    return torch.linalg.cholesky(K)
+
+
+def _batch_lowrank_logdet(W, D, capacitance_tril):
+    r"""
+    Uses "matrix determinant lemma"::
+        log|W @ W.T + D| = log|C| + log|D|,
+    where :math:`C` is the capacitance matrix :math:`I + W.T @ inv(D) @ W`, to compute
+    the log determinant.
+    """
+    return 2 * capacitance_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) + D.log().sum(
+        -1
+    )
+
+
+def _batch_lowrank_mahalanobis(W, D, x, capacitance_tril):
+    r"""
+    Uses "Woodbury matrix identity"::
+        inv(W @ W.T + D) = inv(D) - inv(D) @ W @ inv(C) @ W.T @ inv(D),
+    where :math:`C` is the capacitance matrix :math:`I + W.T @ inv(D) @ W`, to compute the squared
+    Mahalanobis distance :math:`x.T @ inv(W @ W.T + D) @ x`.
+    """
+    Wt_Dinv = W.mT / D.unsqueeze(-2)
+    Wt_Dinv_x = _batch_mv(Wt_Dinv, x)
+    mahalanobis_term1 = (x.pow(2) / D).sum(-1)
+    mahalanobis_term2 = _batch_mahalanobis(capacitance_tril, Wt_Dinv_x)
+    return mahalanobis_term1 - mahalanobis_term2
+
+
+class LowRankMultivariateNormal(Distribution):
+    r"""
+    Creates a multivariate normal distribution with covariance matrix having a low-rank form
+    parameterized by :attr:`cov_factor` and :attr:`cov_diag`::
+
+        covariance_matrix = cov_factor @ cov_factor.T + cov_diag
+
+    Example:
+        >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = LowRankMultivariateNormal(torch.zeros(2), torch.tensor([[1.], [0.]]), torch.ones(2))
+        >>> m.sample()  # normally distributed with mean=`[0,0]`, cov_factor=`[[1],[0]]`, cov_diag=`[1,1]`
+        tensor([-0.2102, -0.5429])
+
+    Args:
+        loc (Tensor): mean of the distribution with shape `batch_shape + event_shape`
+        cov_factor (Tensor): factor part of low-rank form of covariance matrix with shape
+            `batch_shape + event_shape + (rank,)`
+        cov_diag (Tensor): diagonal part of low-rank form of covariance matrix with shape
+            `batch_shape + event_shape`
+
+    Note:
+        The computation for determinant and inverse of covariance matrix is avoided when
+        `cov_factor.shape[1] << cov_factor.shape[0]` thanks to `Woodbury matrix identity
+        <https://en.wikipedia.org/wiki/Woodbury_matrix_identity>`_ and
+        `matrix determinant lemma <https://en.wikipedia.org/wiki/Matrix_determinant_lemma>`_.
+        Thanks to these formulas, we just need to compute the determinant and inverse of
+        the small size "capacitance" matrix::
+
+            capacitance = I + cov_factor.T @ inv(cov_diag) @ cov_factor
+    """
+    arg_constraints = {
+        "loc": constraints.real_vector,
+        "cov_factor": constraints.independent(constraints.real, 2),
+        "cov_diag": constraints.independent(constraints.positive, 1),
+    }
+    support = constraints.real_vector
+    has_rsample = True
+
+    def __init__(self, loc, cov_factor, cov_diag, validate_args=None):
+        if loc.dim() < 1:
+            raise ValueError("loc must be at least one-dimensional.")
+        event_shape = loc.shape[-1:]
+        if cov_factor.dim() < 2:
+            raise ValueError(
+                "cov_factor must be at least two-dimensional, "
+                "with optional leading batch dimensions"
+            )
+        if cov_factor.shape[-2:-1] != event_shape:
+            raise ValueError(
+                f"cov_factor must be a batch of matrices with shape {event_shape[0]} x m"
+            )
+        if cov_diag.shape[-1:] != event_shape:
+            raise ValueError(
+                f"cov_diag must be a batch of vectors with shape {event_shape}"
+            )
+
+        loc_ = loc.unsqueeze(-1)
+        cov_diag_ = cov_diag.unsqueeze(-1)
+        try:
+            loc_, self.cov_factor, cov_diag_ = torch.broadcast_tensors(
+                loc_, cov_factor, cov_diag_
+            )
+        except RuntimeError as e:
+            raise ValueError(
+                f"Incompatible batch shapes: loc {loc.shape}, cov_factor {cov_factor.shape}, cov_diag {cov_diag.shape}"
+            ) from e
+        self.loc = loc_[..., 0]
+        self.cov_diag = cov_diag_[..., 0]
+        batch_shape = self.loc.shape[:-1]
+
+        self._unbroadcasted_cov_factor = cov_factor
+        self._unbroadcasted_cov_diag = cov_diag
+        self._capacitance_tril = _batch_capacitance_tril(cov_factor, cov_diag)
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(LowRankMultivariateNormal, _instance)
+        batch_shape = torch.Size(batch_shape)
+        loc_shape = batch_shape + self.event_shape
+        new.loc = self.loc.expand(loc_shape)
+        new.cov_diag = self.cov_diag.expand(loc_shape)
+        new.cov_factor = self.cov_factor.expand(loc_shape + self.cov_factor.shape[-1:])
+        new._unbroadcasted_cov_factor = self._unbroadcasted_cov_factor
+        new._unbroadcasted_cov_diag = self._unbroadcasted_cov_diag
+        new._capacitance_tril = self._capacitance_tril
+        super(LowRankMultivariateNormal, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def mean(self):
+        return self.loc
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @lazy_property
+    def variance(self):
+        return (
+            self._unbroadcasted_cov_factor.pow(2).sum(-1) + self._unbroadcasted_cov_diag
+        ).expand(self._batch_shape + self._event_shape)
+
+    @lazy_property
+    def scale_tril(self):
+        # The following identity is used to increase the numerically computation stability
+        # for Cholesky decomposition (see http://www.gaussianprocess.org/gpml/, Section 3.4.3):
+        #     W @ W.T + D = D1/2 @ (I + D-1/2 @ W @ W.T @ D-1/2) @ D1/2
+        # The matrix "I + D-1/2 @ W @ W.T @ D-1/2" has eigenvalues bounded from below by 1,
+        # hence it is well-conditioned and safe to take Cholesky decomposition.
+        n = self._event_shape[0]
+        cov_diag_sqrt_unsqueeze = self._unbroadcasted_cov_diag.sqrt().unsqueeze(-1)
+        Dinvsqrt_W = self._unbroadcasted_cov_factor / cov_diag_sqrt_unsqueeze
+        K = torch.matmul(Dinvsqrt_W, Dinvsqrt_W.mT).contiguous()
+        K.view(-1, n * n)[:, :: n + 1] += 1  # add identity matrix to K
+        scale_tril = cov_diag_sqrt_unsqueeze * torch.linalg.cholesky(K)
+        return scale_tril.expand(
+            self._batch_shape + self._event_shape + self._event_shape
+        )
+
+    @lazy_property
+    def covariance_matrix(self):
+        covariance_matrix = torch.matmul(
+            self._unbroadcasted_cov_factor, self._unbroadcasted_cov_factor.mT
+        ) + torch.diag_embed(self._unbroadcasted_cov_diag)
+        return covariance_matrix.expand(
+            self._batch_shape + self._event_shape + self._event_shape
+        )
+
+    @lazy_property
+    def precision_matrix(self):
+        # We use "Woodbury matrix identity" to take advantage of low rank form::
+        #     inv(W @ W.T + D) = inv(D) - inv(D) @ W @ inv(C) @ W.T @ inv(D)
+        # where :math:`C` is the capacitance matrix.
+        Wt_Dinv = (
+            self._unbroadcasted_cov_factor.mT
+            / self._unbroadcasted_cov_diag.unsqueeze(-2)
+        )
+        A = torch.linalg.solve_triangular(self._capacitance_tril, Wt_Dinv, upper=False)
+        precision_matrix = (
+            torch.diag_embed(self._unbroadcasted_cov_diag.reciprocal()) - A.mT @ A
+        )
+        return precision_matrix.expand(
+            self._batch_shape + self._event_shape + self._event_shape
+        )
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        W_shape = shape[:-1] + self.cov_factor.shape[-1:]
+        eps_W = _standard_normal(W_shape, dtype=self.loc.dtype, device=self.loc.device)
+        eps_D = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
+        return (
+            self.loc
+            + _batch_mv(self._unbroadcasted_cov_factor, eps_W)
+            + self._unbroadcasted_cov_diag.sqrt() * eps_D
+        )
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        diff = value - self.loc
+        M = _batch_lowrank_mahalanobis(
+            self._unbroadcasted_cov_factor,
+            self._unbroadcasted_cov_diag,
+            diff,
+            self._capacitance_tril,
+        )
+        log_det = _batch_lowrank_logdet(
+            self._unbroadcasted_cov_factor,
+            self._unbroadcasted_cov_diag,
+            self._capacitance_tril,
+        )
+        return -0.5 * (self._event_shape[0] * math.log(2 * math.pi) + log_det + M)
+
+    def entropy(self):
+        log_det = _batch_lowrank_logdet(
+            self._unbroadcasted_cov_factor,
+            self._unbroadcasted_cov_diag,
+            self._capacitance_tril,
+        )
+        H = 0.5 * (self._event_shape[0] * (1.0 + math.log(2 * math.pi)) + log_det)
+        if len(self._batch_shape) == 0:
+            return H
+        else:
+            return H.expand(self._batch_shape)

rl/Lib/site-packages/torch/distributions/mixture_same_family.py

@@ -0,0 +1,216 @@
+# mypy: allow-untyped-defs
+from typing import Dict
+
+import torch
+from torch.distributions import Categorical, constraints
+from torch.distributions.distribution import Distribution
+
+
+__all__ = ["MixtureSameFamily"]
+
+
+class MixtureSameFamily(Distribution):
+    r"""
+    The `MixtureSameFamily` distribution implements a (batch of) mixture
+    distribution where all component are from different parameterizations of
+    the same distribution type. It is parameterized by a `Categorical`
+    "selecting distribution" (over `k` component) and a component
+    distribution, i.e., a `Distribution` with a rightmost batch shape
+    (equal to `[k]`) which indexes each (batch of) component.
+
+    Examples::
+
+        >>> # xdoctest: +SKIP("undefined vars")
+        >>> # Construct Gaussian Mixture Model in 1D consisting of 5 equally
+        >>> # weighted normal distributions
+        >>> mix = D.Categorical(torch.ones(5,))
+        >>> comp = D.Normal(torch.randn(5,), torch.rand(5,))
+        >>> gmm = MixtureSameFamily(mix, comp)
+
+        >>> # Construct Gaussian Mixture Model in 2D consisting of 5 equally
+        >>> # weighted bivariate normal distributions
+        >>> mix = D.Categorical(torch.ones(5,))
+        >>> comp = D.Independent(D.Normal(
+        ...          torch.randn(5,2), torch.rand(5,2)), 1)
+        >>> gmm = MixtureSameFamily(mix, comp)
+
+        >>> # Construct a batch of 3 Gaussian Mixture Models in 2D each
+        >>> # consisting of 5 random weighted bivariate normal distributions
+        >>> mix = D.Categorical(torch.rand(3,5))
+        >>> comp = D.Independent(D.Normal(
+        ...         torch.randn(3,5,2), torch.rand(3,5,2)), 1)
+        >>> gmm = MixtureSameFamily(mix, comp)
+
+    Args:
+        mixture_distribution: `torch.distributions.Categorical`-like
+            instance. Manages the probability of selecting component.
+            The number of categories must match the rightmost batch
+            dimension of the `component_distribution`. Must have either
+            scalar `batch_shape` or `batch_shape` matching
+            `component_distribution.batch_shape[:-1]`
+        component_distribution: `torch.distributions.Distribution`-like
+            instance. Right-most batch dimension indexes component.
+    """
+    arg_constraints: Dict[str, constraints.Constraint] = {}
+    has_rsample = False
+
+    def __init__(
+        self, mixture_distribution, component_distribution, validate_args=None
+    ):
+        self._mixture_distribution = mixture_distribution
+        self._component_distribution = component_distribution
+
+        if not isinstance(self._mixture_distribution, Categorical):
+            raise ValueError(
+                " The Mixture distribution needs to be an "
+                " instance of torch.distributions.Categorical"
+            )
+
+        if not isinstance(self._component_distribution, Distribution):
+            raise ValueError(
+                "The Component distribution need to be an "
+                "instance of torch.distributions.Distribution"
+            )
+
+        # Check that batch size matches
+        mdbs = self._mixture_distribution.batch_shape
+        cdbs = self._component_distribution.batch_shape[:-1]
+        for size1, size2 in zip(reversed(mdbs), reversed(cdbs)):
+            if size1 != 1 and size2 != 1 and size1 != size2:
+                raise ValueError(
+                    f"`mixture_distribution.batch_shape` ({mdbs}) is not "
+                    "compatible with `component_distribution."
+                    f"batch_shape`({cdbs})"
+                )
+
+        # Check that the number of mixture component matches
+        km = self._mixture_distribution.logits.shape[-1]
+        kc = self._component_distribution.batch_shape[-1]
+        if km is not None and kc is not None and km != kc:
+            raise ValueError(
+                f"`mixture_distribution component` ({km}) does not"
+                " equal `component_distribution.batch_shape[-1]`"
+                f" ({kc})"
+            )
+        self._num_component = km
+
+        event_shape = self._component_distribution.event_shape
+        self._event_ndims = len(event_shape)
+        super().__init__(
+            batch_shape=cdbs, event_shape=event_shape, validate_args=validate_args
+        )
+
+    def expand(self, batch_shape, _instance=None):
+        batch_shape = torch.Size(batch_shape)
+        batch_shape_comp = batch_shape + (self._num_component,)
+        new = self._get_checked_instance(MixtureSameFamily, _instance)
+        new._component_distribution = self._component_distribution.expand(
+            batch_shape_comp
+        )
+        new._mixture_distribution = self._mixture_distribution.expand(batch_shape)
+        new._num_component = self._num_component
+        new._event_ndims = self._event_ndims
+        event_shape = new._component_distribution.event_shape
+        super(MixtureSameFamily, new).__init__(
+            batch_shape=batch_shape, event_shape=event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    @constraints.dependent_property
+    def support(self):
+        # FIXME this may have the wrong shape when support contains batched
+        # parameters
+        return self._component_distribution.support
+
+    @property
+    def mixture_distribution(self):
+        return self._mixture_distribution
+
+    @property
+    def component_distribution(self):
+        return self._component_distribution
+
+    @property
+    def mean(self):
+        probs = self._pad_mixture_dimensions(self.mixture_distribution.probs)
+        return torch.sum(
+            probs * self.component_distribution.mean, dim=-1 - self._event_ndims
+        )  # [B, E]
+
+    @property
+    def variance(self):
+        # Law of total variance: Var(Y) = E[Var(Y|X)] + Var(E[Y|X])
+        probs = self._pad_mixture_dimensions(self.mixture_distribution.probs)
+        mean_cond_var = torch.sum(
+            probs * self.component_distribution.variance, dim=-1 - self._event_ndims
+        )
+        var_cond_mean = torch.sum(
+            probs * (self.component_distribution.mean - self._pad(self.mean)).pow(2.0),
+            dim=-1 - self._event_ndims,
+        )
+        return mean_cond_var + var_cond_mean
+
+    def cdf(self, x):
+        x = self._pad(x)
+        cdf_x = self.component_distribution.cdf(x)
+        mix_prob = self.mixture_distribution.probs
+
+        return torch.sum(cdf_x * mix_prob, dim=-1)
+
+    def log_prob(self, x):
+        if self._validate_args:
+            self._validate_sample(x)
+        x = self._pad(x)
+        log_prob_x = self.component_distribution.log_prob(x)  # [S, B, k]
+        log_mix_prob = torch.log_softmax(
+            self.mixture_distribution.logits, dim=-1
+        )  # [B, k]
+        return torch.logsumexp(log_prob_x + log_mix_prob, dim=-1)  # [S, B]
+
+    def sample(self, sample_shape=torch.Size()):
+        with torch.no_grad():
+            sample_len = len(sample_shape)
+            batch_len = len(self.batch_shape)
+            gather_dim = sample_len + batch_len
+            es = self.event_shape
+
+            # mixture samples [n, B]
+            mix_sample = self.mixture_distribution.sample(sample_shape)
+            mix_shape = mix_sample.shape
+
+            # component samples [n, B, k, E]
+            comp_samples = self.component_distribution.sample(sample_shape)
+
+            # Gather along the k dimension
+            mix_sample_r = mix_sample.reshape(
+                mix_shape + torch.Size([1] * (len(es) + 1))
+            )
+            mix_sample_r = mix_sample_r.repeat(
+                torch.Size([1] * len(mix_shape)) + torch.Size([1]) + es
+            )
+
+            samples = torch.gather(comp_samples, gather_dim, mix_sample_r)
+            return samples.squeeze(gather_dim)
+
+    def _pad(self, x):
+        return x.unsqueeze(-1 - self._event_ndims)
+
+    def _pad_mixture_dimensions(self, x):
+        dist_batch_ndims = len(self.batch_shape)
+        cat_batch_ndims = len(self.mixture_distribution.batch_shape)
+        pad_ndims = 0 if cat_batch_ndims == 1 else dist_batch_ndims - cat_batch_ndims
+        xs = x.shape
+        x = x.reshape(
+            xs[:-1]
+            + torch.Size(pad_ndims * [1])
+            + xs[-1:]
+            + torch.Size(self._event_ndims * [1])
+        )
+        return x
+
+    def __repr__(self):
+        args_string = (
+            f"\n  {self.mixture_distribution},\n  {self.component_distribution}"
+        )
+        return "MixtureSameFamily" + "(" + args_string + ")"

rl/Lib/site-packages/torch/distributions/multinomial.py

@@ -0,0 +1,137 @@
+# mypy: allow-untyped-defs
+import torch
+from torch import inf
+from torch.distributions import Categorical, constraints
+from torch.distributions.binomial import Binomial
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import broadcast_all
+
+
+__all__ = ["Multinomial"]
+
+
+class Multinomial(Distribution):
+    r"""
+    Creates a Multinomial distribution parameterized by :attr:`total_count` and
+    either :attr:`probs` or :attr:`logits` (but not both). The innermost dimension of
+    :attr:`probs` indexes over categories. All other dimensions index over batches.
+
+    Note that :attr:`total_count` need not be specified if only :meth:`log_prob` is
+    called (see example below)
+
+    .. note:: The `probs` argument must be non-negative, finite and have a non-zero sum,
+              and it will be normalized to sum to 1 along the last dimension. :attr:`probs`
+              will return this normalized value.
+              The `logits` argument will be interpreted as unnormalized log probabilities
+              and can therefore be any real number. It will likewise be normalized so that
+              the resulting probabilities sum to 1 along the last dimension. :attr:`logits`
+              will return this normalized value.
+
+    -   :meth:`sample` requires a single shared `total_count` for all
+        parameters and samples.
+    -   :meth:`log_prob` allows different `total_count` for each parameter and
+        sample.
+
+    Example::
+
+        >>> # xdoctest: +SKIP("FIXME: found invalid values")
+        >>> m = Multinomial(100, torch.tensor([ 1., 1., 1., 1.]))
+        >>> x = m.sample()  # equal probability of 0, 1, 2, 3
+        tensor([ 21.,  24.,  30.,  25.])
+
+        >>> Multinomial(probs=torch.tensor([1., 1., 1., 1.])).log_prob(x)
+        tensor([-4.1338])
+
+    Args:
+        total_count (int): number of trials
+        probs (Tensor): event probabilities
+        logits (Tensor): event log probabilities (unnormalized)
+    """
+    arg_constraints = {"probs": constraints.simplex, "logits": constraints.real_vector}
+    total_count: int
+
+    @property
+    def mean(self):
+        return self.probs * self.total_count
+
+    @property
+    def variance(self):
+        return self.total_count * self.probs * (1 - self.probs)
+
+    def __init__(self, total_count=1, probs=None, logits=None, validate_args=None):
+        if not isinstance(total_count, int):
+            raise NotImplementedError("inhomogeneous total_count is not supported")
+        self.total_count = total_count
+        self._categorical = Categorical(probs=probs, logits=logits)
+        self._binomial = Binomial(total_count=total_count, probs=self.probs)
+        batch_shape = self._categorical.batch_shape
+        event_shape = self._categorical.param_shape[-1:]
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Multinomial, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.total_count = self.total_count
+        new._categorical = self._categorical.expand(batch_shape)
+        super(Multinomial, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._categorical._new(*args, **kwargs)
+
+    @constraints.dependent_property(is_discrete=True, event_dim=1)
+    def support(self):
+        return constraints.multinomial(self.total_count)
+
+    @property
+    def logits(self):
+        return self._categorical.logits
+
+    @property
+    def probs(self):
+        return self._categorical.probs
+
+    @property
+    def param_shape(self):
+        return self._categorical.param_shape
+
+    def sample(self, sample_shape=torch.Size()):
+        sample_shape = torch.Size(sample_shape)
+        samples = self._categorical.sample(
+            torch.Size((self.total_count,)) + sample_shape
+        )
+        # samples.shape is (total_count, sample_shape, batch_shape), need to change it to
+        # (sample_shape, batch_shape, total_count)
+        shifted_idx = list(range(samples.dim()))
+        shifted_idx.append(shifted_idx.pop(0))
+        samples = samples.permute(*shifted_idx)
+        counts = samples.new(self._extended_shape(sample_shape)).zero_()
+        counts.scatter_add_(-1, samples, torch.ones_like(samples))
+        return counts.type_as(self.probs)
+
+    def entropy(self):
+        n = torch.tensor(self.total_count)
+
+        cat_entropy = self._categorical.entropy()
+        term1 = n * cat_entropy - torch.lgamma(n + 1)
+
+        support = self._binomial.enumerate_support(expand=False)[1:]
+        binomial_probs = torch.exp(self._binomial.log_prob(support))
+        weights = torch.lgamma(support + 1)
+        term2 = (binomial_probs * weights).sum([0, -1])
+
+        return term1 + term2
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        logits, value = broadcast_all(self.logits, value)
+        logits = logits.clone(memory_format=torch.contiguous_format)
+        log_factorial_n = torch.lgamma(value.sum(-1) + 1)
+        log_factorial_xs = torch.lgamma(value + 1).sum(-1)
+        logits[(value == 0) & (logits == -inf)] = 0
+        log_powers = (logits * value).sum(-1)
+        return log_factorial_n - log_factorial_xs + log_powers

rl/Lib/site-packages/torch/distributions/multivariate_normal.py

@@ -0,0 +1,265 @@
+# mypy: allow-untyped-defs
+import math
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import _standard_normal, lazy_property
+from torch.types import _size
+
+
+__all__ = ["MultivariateNormal"]
+
+
+def _batch_mv(bmat, bvec):
+    r"""
+    Performs a batched matrix-vector product, with compatible but different batch shapes.
+
+    This function takes as input `bmat`, containing :math:`n \times n` matrices, and
+    `bvec`, containing length :math:`n` vectors.
+
+    Both `bmat` and `bvec` may have any number of leading dimensions, which correspond
+    to a batch shape. They are not necessarily assumed to have the same batch shape,
+    just ones which can be broadcasted.
+    """
+    return torch.matmul(bmat, bvec.unsqueeze(-1)).squeeze(-1)
+
+
+def _batch_mahalanobis(bL, bx):
+    r"""
+    Computes the squared Mahalanobis distance :math:`\mathbf{x}^\top\mathbf{M}^{-1}\mathbf{x}`
+    for a factored :math:`\mathbf{M} = \mathbf{L}\mathbf{L}^\top`.
+
+    Accepts batches for both bL and bx. They are not necessarily assumed to have the same batch
+    shape, but `bL` one should be able to broadcasted to `bx` one.
+    """
+    n = bx.size(-1)
+    bx_batch_shape = bx.shape[:-1]
+
+    # Assume that bL.shape = (i, 1, n, n), bx.shape = (..., i, j, n),
+    # we are going to make bx have shape (..., 1, j,  i, 1, n) to apply batched tri.solve
+    bx_batch_dims = len(bx_batch_shape)
+    bL_batch_dims = bL.dim() - 2
+    outer_batch_dims = bx_batch_dims - bL_batch_dims
+    old_batch_dims = outer_batch_dims + bL_batch_dims
+    new_batch_dims = outer_batch_dims + 2 * bL_batch_dims
+    # Reshape bx with the shape (..., 1, i, j, 1, n)
+    bx_new_shape = bx.shape[:outer_batch_dims]
+    for sL, sx in zip(bL.shape[:-2], bx.shape[outer_batch_dims:-1]):
+        bx_new_shape += (sx // sL, sL)
+    bx_new_shape += (n,)
+    bx = bx.reshape(bx_new_shape)
+    # Permute bx to make it have shape (..., 1, j, i, 1, n)
+    permute_dims = (
+        list(range(outer_batch_dims))
+        + list(range(outer_batch_dims, new_batch_dims, 2))
+        + list(range(outer_batch_dims + 1, new_batch_dims, 2))
+        + [new_batch_dims]
+    )
+    bx = bx.permute(permute_dims)
+
+    flat_L = bL.reshape(-1, n, n)  # shape = b x n x n
+    flat_x = bx.reshape(-1, flat_L.size(0), n)  # shape = c x b x n
+    flat_x_swap = flat_x.permute(1, 2, 0)  # shape = b x n x c
+    M_swap = (
+        torch.linalg.solve_triangular(flat_L, flat_x_swap, upper=False).pow(2).sum(-2)
+    )  # shape = b x c
+    M = M_swap.t()  # shape = c x b
+
+    # Now we revert the above reshape and permute operators.
+    permuted_M = M.reshape(bx.shape[:-1])  # shape = (..., 1, j, i, 1)
+    permute_inv_dims = list(range(outer_batch_dims))
+    for i in range(bL_batch_dims):
+        permute_inv_dims += [outer_batch_dims + i, old_batch_dims + i]
+    reshaped_M = permuted_M.permute(permute_inv_dims)  # shape = (..., 1, i, j, 1)
+    return reshaped_M.reshape(bx_batch_shape)
+
+
+def _precision_to_scale_tril(P):
+    # Ref: https://nbviewer.jupyter.org/gist/fehiepsi/5ef8e09e61604f10607380467eb82006#Precision-to-scale_tril
+    Lf = torch.linalg.cholesky(torch.flip(P, (-2, -1)))
+    L_inv = torch.transpose(torch.flip(Lf, (-2, -1)), -2, -1)
+    Id = torch.eye(P.shape[-1], dtype=P.dtype, device=P.device)
+    L = torch.linalg.solve_triangular(L_inv, Id, upper=False)
+    return L
+
+
+class MultivariateNormal(Distribution):
+    r"""
+    Creates a multivariate normal (also called Gaussian) distribution
+    parameterized by a mean vector and a covariance matrix.
+
+    The multivariate normal distribution can be parameterized either
+    in terms of a positive definite covariance matrix :math:`\mathbf{\Sigma}`
+    or a positive definite precision matrix :math:`\mathbf{\Sigma}^{-1}`
+    or a lower-triangular matrix :math:`\mathbf{L}` with positive-valued
+    diagonal entries, such that
+    :math:`\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top`. This triangular matrix
+    can be obtained via e.g. Cholesky decomposition of the covariance.
+
+    Example:
+
+        >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = MultivariateNormal(torch.zeros(2), torch.eye(2))
+        >>> m.sample()  # normally distributed with mean=`[0,0]` and covariance_matrix=`I`
+        tensor([-0.2102, -0.5429])
+
+    Args:
+        loc (Tensor): mean of the distribution
+        covariance_matrix (Tensor): positive-definite covariance matrix
+        precision_matrix (Tensor): positive-definite precision matrix
+        scale_tril (Tensor): lower-triangular factor of covariance, with positive-valued diagonal
+
+    Note:
+        Only one of :attr:`covariance_matrix` or :attr:`precision_matrix` or
+        :attr:`scale_tril` can be specified.
+
+        Using :attr:`scale_tril` will be more efficient: all computations internally
+        are based on :attr:`scale_tril`. If :attr:`covariance_matrix` or
+        :attr:`precision_matrix` is passed instead, it is only used to compute
+        the corresponding lower triangular matrices using a Cholesky decomposition.
+    """
+    arg_constraints = {
+        "loc": constraints.real_vector,
+        "covariance_matrix": constraints.positive_definite,
+        "precision_matrix": constraints.positive_definite,
+        "scale_tril": constraints.lower_cholesky,
+    }
+    support = constraints.real_vector
+    has_rsample = True
+
+    def __init__(
+        self,
+        loc,
+        covariance_matrix=None,
+        precision_matrix=None,
+        scale_tril=None,
+        validate_args=None,
+    ):
+        if loc.dim() < 1:
+            raise ValueError("loc must be at least one-dimensional.")
+        if (covariance_matrix is not None) + (scale_tril is not None) + (
+            precision_matrix is not None
+        ) != 1:
+            raise ValueError(
+                "Exactly one of covariance_matrix or precision_matrix or scale_tril may be specified."
+            )
+
+        if scale_tril is not None:
+            if scale_tril.dim() < 2:
+                raise ValueError(
+                    "scale_tril matrix must be at least two-dimensional, "
+                    "with optional leading batch dimensions"
+                )
+            batch_shape = torch.broadcast_shapes(scale_tril.shape[:-2], loc.shape[:-1])
+            self.scale_tril = scale_tril.expand(batch_shape + (-1, -1))
+        elif covariance_matrix is not None:
+            if covariance_matrix.dim() < 2:
+                raise ValueError(
+                    "covariance_matrix must be at least two-dimensional, "
+                    "with optional leading batch dimensions"
+                )
+            batch_shape = torch.broadcast_shapes(
+                covariance_matrix.shape[:-2], loc.shape[:-1]
+            )
+            self.covariance_matrix = covariance_matrix.expand(batch_shape + (-1, -1))
+        else:
+            if precision_matrix.dim() < 2:
+                raise ValueError(
+                    "precision_matrix must be at least two-dimensional, "
+                    "with optional leading batch dimensions"
+                )
+            batch_shape = torch.broadcast_shapes(
+                precision_matrix.shape[:-2], loc.shape[:-1]
+            )
+            self.precision_matrix = precision_matrix.expand(batch_shape + (-1, -1))
+        self.loc = loc.expand(batch_shape + (-1,))
+
+        event_shape = self.loc.shape[-1:]
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+        if scale_tril is not None:
+            self._unbroadcasted_scale_tril = scale_tril
+        elif covariance_matrix is not None:
+            self._unbroadcasted_scale_tril = torch.linalg.cholesky(covariance_matrix)
+        else:  # precision_matrix is not None
+            self._unbroadcasted_scale_tril = _precision_to_scale_tril(precision_matrix)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(MultivariateNormal, _instance)
+        batch_shape = torch.Size(batch_shape)
+        loc_shape = batch_shape + self.event_shape
+        cov_shape = batch_shape + self.event_shape + self.event_shape
+        new.loc = self.loc.expand(loc_shape)
+        new._unbroadcasted_scale_tril = self._unbroadcasted_scale_tril
+        if "covariance_matrix" in self.__dict__:
+            new.covariance_matrix = self.covariance_matrix.expand(cov_shape)
+        if "scale_tril" in self.__dict__:
+            new.scale_tril = self.scale_tril.expand(cov_shape)
+        if "precision_matrix" in self.__dict__:
+            new.precision_matrix = self.precision_matrix.expand(cov_shape)
+        super(MultivariateNormal, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    @lazy_property
+    def scale_tril(self):
+        return self._unbroadcasted_scale_tril.expand(
+            self._batch_shape + self._event_shape + self._event_shape
+        )
+
+    @lazy_property
+    def covariance_matrix(self):
+        return torch.matmul(
+            self._unbroadcasted_scale_tril, self._unbroadcasted_scale_tril.mT
+        ).expand(self._batch_shape + self._event_shape + self._event_shape)
+
+    @lazy_property
+    def precision_matrix(self):
+        return torch.cholesky_inverse(self._unbroadcasted_scale_tril).expand(
+            self._batch_shape + self._event_shape + self._event_shape
+        )
+
+    @property
+    def mean(self):
+        return self.loc
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @property
+    def variance(self):
+        return (
+            self._unbroadcasted_scale_tril.pow(2)
+            .sum(-1)
+            .expand(self._batch_shape + self._event_shape)
+        )
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        eps = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
+        return self.loc + _batch_mv(self._unbroadcasted_scale_tril, eps)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        diff = value - self.loc
+        M = _batch_mahalanobis(self._unbroadcasted_scale_tril, diff)
+        half_log_det = (
+            self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)
+        )
+        return -0.5 * (self._event_shape[0] * math.log(2 * math.pi) + M) - half_log_det
+
+    def entropy(self):
+        half_log_det = (
+            self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1)
+        )
+        H = 0.5 * self._event_shape[0] * (1.0 + math.log(2 * math.pi)) + half_log_det
+        if len(self._batch_shape) == 0:
+            return H
+        else:
+            return H.expand(self._batch_shape)

rl/Lib/site-packages/torch/distributions/negative_binomial.py

@@ -0,0 +1,135 @@
+# mypy: allow-untyped-defs
+import torch
+import torch.nn.functional as F
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import (
+    broadcast_all,
+    lazy_property,
+    logits_to_probs,
+    probs_to_logits,
+)
+
+
+__all__ = ["NegativeBinomial"]
+
+
+class NegativeBinomial(Distribution):
+    r"""
+    Creates a Negative Binomial distribution, i.e. distribution
+    of the number of successful independent and identical Bernoulli trials
+    before :attr:`total_count` failures are achieved. The probability
+    of success of each Bernoulli trial is :attr:`probs`.
+
+    Args:
+        total_count (float or Tensor): non-negative number of negative Bernoulli
+            trials to stop, although the distribution is still valid for real
+            valued count
+        probs (Tensor): Event probabilities of success in the half open interval [0, 1)
+        logits (Tensor): Event log-odds for probabilities of success
+    """
+    arg_constraints = {
+        "total_count": constraints.greater_than_eq(0),
+        "probs": constraints.half_open_interval(0.0, 1.0),
+        "logits": constraints.real,
+    }
+    support = constraints.nonnegative_integer
+
+    def __init__(self, total_count, probs=None, logits=None, validate_args=None):
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            (
+                self.total_count,
+                self.probs,
+            ) = broadcast_all(total_count, probs)
+            self.total_count = self.total_count.type_as(self.probs)
+        else:
+            (
+                self.total_count,
+                self.logits,
+            ) = broadcast_all(total_count, logits)
+            self.total_count = self.total_count.type_as(self.logits)
+
+        self._param = self.probs if probs is not None else self.logits
+        batch_shape = self._param.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(NegativeBinomial, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.total_count = self.total_count.expand(batch_shape)
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(batch_shape)
+            new._param = new.probs
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(batch_shape)
+            new._param = new.logits
+        super(NegativeBinomial, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._param.new(*args, **kwargs)
+
+    @property
+    def mean(self):
+        return self.total_count * torch.exp(self.logits)
+
+    @property
+    def mode(self):
+        return ((self.total_count - 1) * self.logits.exp()).floor().clamp(min=0.0)
+
+    @property
+    def variance(self):
+        return self.mean / torch.sigmoid(-self.logits)
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs, is_binary=True)
+
+    @lazy_property
+    def probs(self):
+        return logits_to_probs(self.logits, is_binary=True)
+
+    @property
+    def param_shape(self):
+        return self._param.size()
+
+    @lazy_property
+    def _gamma(self):
+        # Note we avoid validating because self.total_count can be zero.
+        return torch.distributions.Gamma(
+            concentration=self.total_count,
+            rate=torch.exp(-self.logits),
+            validate_args=False,
+        )
+
+    def sample(self, sample_shape=torch.Size()):
+        with torch.no_grad():
+            rate = self._gamma.sample(sample_shape=sample_shape)
+            return torch.poisson(rate)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+
+        log_unnormalized_prob = self.total_count * F.logsigmoid(
+            -self.logits
+        ) + value * F.logsigmoid(self.logits)
+
+        log_normalization = (
+            -torch.lgamma(self.total_count + value)
+            + torch.lgamma(1.0 + value)
+            + torch.lgamma(self.total_count)
+        )
+        # The case self.total_count == 0 and value == 0 has probability 1 but
+        # lgamma(0) is infinite. Handle this case separately using a function
+        # that does not modify tensors in place to allow Jit compilation.
+        log_normalization = log_normalization.masked_fill(
+            self.total_count + value == 0.0, 0.0
+        )
+
+        return log_unnormalized_prob - log_normalization

rl/Lib/site-packages/torch/distributions/normal.py

@@ -0,0 +1,112 @@
+# mypy: allow-untyped-defs
+import math
+from numbers import Number, Real
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import _standard_normal, broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Normal"]
+
+
+class Normal(ExponentialFamily):
+    r"""
+    Creates a normal (also called Gaussian) distribution parameterized by
+    :attr:`loc` and :attr:`scale`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Normal(torch.tensor([0.0]), torch.tensor([1.0]))
+        >>> m.sample()  # normally distributed with loc=0 and scale=1
+        tensor([ 0.1046])
+
+    Args:
+        loc (float or Tensor): mean of the distribution (often referred to as mu)
+        scale (float or Tensor): standard deviation of the distribution
+            (often referred to as sigma)
+    """
+    arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
+    support = constraints.real
+    has_rsample = True
+    _mean_carrier_measure = 0
+
+    @property
+    def mean(self):
+        return self.loc
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @property
+    def stddev(self):
+        return self.scale
+
+    @property
+    def variance(self):
+        return self.stddev.pow(2)
+
+    def __init__(self, loc, scale, validate_args=None):
+        self.loc, self.scale = broadcast_all(loc, scale)
+        if isinstance(loc, Number) and isinstance(scale, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.loc.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Normal, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.loc = self.loc.expand(batch_shape)
+        new.scale = self.scale.expand(batch_shape)
+        super(Normal, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def sample(self, sample_shape=torch.Size()):
+        shape = self._extended_shape(sample_shape)
+        with torch.no_grad():
+            return torch.normal(self.loc.expand(shape), self.scale.expand(shape))
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        eps = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
+        return self.loc + eps * self.scale
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        # compute the variance
+        var = self.scale**2
+        log_scale = (
+            math.log(self.scale) if isinstance(self.scale, Real) else self.scale.log()
+        )
+        return (
+            -((value - self.loc) ** 2) / (2 * var)
+            - log_scale
+            - math.log(math.sqrt(2 * math.pi))
+        )
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        return 0.5 * (
+            1 + torch.erf((value - self.loc) * self.scale.reciprocal() / math.sqrt(2))
+        )
+
+    def icdf(self, value):
+        return self.loc + self.scale * torch.erfinv(2 * value - 1) * math.sqrt(2)
+
+    def entropy(self):
+        return 0.5 + 0.5 * math.log(2 * math.pi) + torch.log(self.scale)
+
+    @property
+    def _natural_params(self):
+        return (self.loc / self.scale.pow(2), -0.5 * self.scale.pow(2).reciprocal())
+
+    def _log_normalizer(self, x, y):
+        return -0.25 * x.pow(2) / y + 0.5 * torch.log(-math.pi / y)

rl/Lib/site-packages/torch/distributions/one_hot_categorical.py

@@ -0,0 +1,132 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.distributions import constraints
+from torch.distributions.categorical import Categorical
+from torch.distributions.distribution import Distribution
+from torch.types import _size
+
+
+__all__ = ["OneHotCategorical", "OneHotCategoricalStraightThrough"]
+
+
+class OneHotCategorical(Distribution):
+    r"""
+    Creates a one-hot categorical distribution parameterized by :attr:`probs` or
+    :attr:`logits`.
+
+    Samples are one-hot coded vectors of size ``probs.size(-1)``.
+
+    .. note:: The `probs` argument must be non-negative, finite and have a non-zero sum,
+              and it will be normalized to sum to 1 along the last dimension. :attr:`probs`
+              will return this normalized value.
+              The `logits` argument will be interpreted as unnormalized log probabilities
+              and can therefore be any real number. It will likewise be normalized so that
+              the resulting probabilities sum to 1 along the last dimension. :attr:`logits`
+              will return this normalized value.
+
+    See also: :func:`torch.distributions.Categorical` for specifications of
+    :attr:`probs` and :attr:`logits`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = OneHotCategorical(torch.tensor([ 0.25, 0.25, 0.25, 0.25 ]))
+        >>> m.sample()  # equal probability of 0, 1, 2, 3
+        tensor([ 0.,  0.,  0.,  1.])
+
+    Args:
+        probs (Tensor): event probabilities
+        logits (Tensor): event log probabilities (unnormalized)
+    """
+    arg_constraints = {"probs": constraints.simplex, "logits": constraints.real_vector}
+    support = constraints.one_hot
+    has_enumerate_support = True
+
+    def __init__(self, probs=None, logits=None, validate_args=None):
+        self._categorical = Categorical(probs, logits)
+        batch_shape = self._categorical.batch_shape
+        event_shape = self._categorical.param_shape[-1:]
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(OneHotCategorical, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new._categorical = self._categorical.expand(batch_shape)
+        super(OneHotCategorical, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._categorical._new(*args, **kwargs)
+
+    @property
+    def _param(self):
+        return self._categorical._param
+
+    @property
+    def probs(self):
+        return self._categorical.probs
+
+    @property
+    def logits(self):
+        return self._categorical.logits
+
+    @property
+    def mean(self):
+        return self._categorical.probs
+
+    @property
+    def mode(self):
+        probs = self._categorical.probs
+        mode = probs.argmax(axis=-1)
+        return torch.nn.functional.one_hot(mode, num_classes=probs.shape[-1]).to(probs)
+
+    @property
+    def variance(self):
+        return self._categorical.probs * (1 - self._categorical.probs)
+
+    @property
+    def param_shape(self):
+        return self._categorical.param_shape
+
+    def sample(self, sample_shape=torch.Size()):
+        sample_shape = torch.Size(sample_shape)
+        probs = self._categorical.probs
+        num_events = self._categorical._num_events
+        indices = self._categorical.sample(sample_shape)
+        return torch.nn.functional.one_hot(indices, num_events).to(probs)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        indices = value.max(-1)[1]
+        return self._categorical.log_prob(indices)
+
+    def entropy(self):
+        return self._categorical.entropy()
+
+    def enumerate_support(self, expand=True):
+        n = self.event_shape[0]
+        values = torch.eye(n, dtype=self._param.dtype, device=self._param.device)
+        values = values.view((n,) + (1,) * len(self.batch_shape) + (n,))
+        if expand:
+            values = values.expand((n,) + self.batch_shape + (n,))
+        return values
+
+
+class OneHotCategoricalStraightThrough(OneHotCategorical):
+    r"""
+    Creates a reparameterizable :class:`OneHotCategorical` distribution based on the straight-
+    through gradient estimator from [1].
+
+    [1] Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation
+    (Bengio et al., 2013)
+    """
+    has_rsample = True
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        samples = self.sample(sample_shape)
+        probs = self._categorical.probs  # cached via @lazy_property
+        return samples + (probs - probs.detach())

rl/Lib/site-packages/torch/distributions/pareto.py

@@ -0,0 +1,62 @@
+# mypy: allow-untyped-defs
+from torch.distributions import constraints
+from torch.distributions.exponential import Exponential
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import AffineTransform, ExpTransform
+from torch.distributions.utils import broadcast_all
+
+
+__all__ = ["Pareto"]
+
+
+class Pareto(TransformedDistribution):
+    r"""
+    Samples from a Pareto Type 1 distribution.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Pareto(torch.tensor([1.0]), torch.tensor([1.0]))
+        >>> m.sample()  # sample from a Pareto distribution with scale=1 and alpha=1
+        tensor([ 1.5623])
+
+    Args:
+        scale (float or Tensor): Scale parameter of the distribution
+        alpha (float or Tensor): Shape parameter of the distribution
+    """
+    arg_constraints = {"alpha": constraints.positive, "scale": constraints.positive}
+
+    def __init__(self, scale, alpha, validate_args=None):
+        self.scale, self.alpha = broadcast_all(scale, alpha)
+        base_dist = Exponential(self.alpha, validate_args=validate_args)
+        transforms = [ExpTransform(), AffineTransform(loc=0, scale=self.scale)]
+        super().__init__(base_dist, transforms, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Pareto, _instance)
+        new.scale = self.scale.expand(batch_shape)
+        new.alpha = self.alpha.expand(batch_shape)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def mean(self):
+        # mean is inf for alpha <= 1
+        a = self.alpha.clamp(min=1)
+        return a * self.scale / (a - 1)
+
+    @property
+    def mode(self):
+        return self.scale
+
+    @property
+    def variance(self):
+        # var is inf for alpha <= 2
+        a = self.alpha.clamp(min=2)
+        return self.scale.pow(2) * a / ((a - 1).pow(2) * (a - 2))
+
+    @constraints.dependent_property(is_discrete=False, event_dim=0)
+    def support(self):
+        return constraints.greater_than_eq(self.scale)
+
+    def entropy(self):
+        return (self.scale / self.alpha).log() + (1 + self.alpha.reciprocal())

rl/Lib/site-packages/torch/distributions/poisson.py

@@ -0,0 +1,79 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.utils import broadcast_all
+
+
+__all__ = ["Poisson"]
+
+
+class Poisson(ExponentialFamily):
+    r"""
+    Creates a Poisson distribution parameterized by :attr:`rate`, the rate parameter.
+
+    Samples are nonnegative integers, with a pmf given by
+
+    .. math::
+      \mathrm{rate}^k \frac{e^{-\mathrm{rate}}}{k!}
+
+    Example::
+
+        >>> # xdoctest: +SKIP("poisson_cpu not implemented for 'Long'")
+        >>> m = Poisson(torch.tensor([4]))
+        >>> m.sample()
+        tensor([ 3.])
+
+    Args:
+        rate (Number, Tensor): the rate parameter
+    """
+    arg_constraints = {"rate": constraints.nonnegative}
+    support = constraints.nonnegative_integer
+
+    @property
+    def mean(self):
+        return self.rate
+
+    @property
+    def mode(self):
+        return self.rate.floor()
+
+    @property
+    def variance(self):
+        return self.rate
+
+    def __init__(self, rate, validate_args=None):
+        (self.rate,) = broadcast_all(rate)
+        if isinstance(rate, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.rate.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Poisson, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.rate = self.rate.expand(batch_shape)
+        super(Poisson, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def sample(self, sample_shape=torch.Size()):
+        shape = self._extended_shape(sample_shape)
+        with torch.no_grad():
+            return torch.poisson(self.rate.expand(shape))
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        rate, value = broadcast_all(self.rate, value)
+        return value.xlogy(rate) - rate - (value + 1).lgamma()
+
+    @property
+    def _natural_params(self):
+        return (torch.log(self.rate),)
+
+    def _log_normalizer(self, x):
+        return torch.exp(x)

rl/Lib/site-packages/torch/distributions/relaxed_bernoulli.py

@@ -0,0 +1,152 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import SigmoidTransform
+from torch.distributions.utils import (
+    broadcast_all,
+    clamp_probs,
+    lazy_property,
+    logits_to_probs,
+    probs_to_logits,
+)
+from torch.types import _size
+
+
+__all__ = ["LogitRelaxedBernoulli", "RelaxedBernoulli"]
+
+
+class LogitRelaxedBernoulli(Distribution):
+    r"""
+    Creates a LogitRelaxedBernoulli distribution parameterized by :attr:`probs`
+    or :attr:`logits` (but not both), which is the logit of a RelaxedBernoulli
+    distribution.
+
+    Samples are logits of values in (0, 1). See [1] for more details.
+
+    Args:
+        temperature (Tensor): relaxation temperature
+        probs (Number, Tensor): the probability of sampling `1`
+        logits (Number, Tensor): the log-odds of sampling `1`
+
+    [1] The Concrete Distribution: A Continuous Relaxation of Discrete Random
+    Variables (Maddison et al., 2017)
+
+    [2] Categorical Reparametrization with Gumbel-Softmax
+    (Jang et al., 2017)
+    """
+    arg_constraints = {"probs": constraints.unit_interval, "logits": constraints.real}
+    support = constraints.real
+
+    def __init__(self, temperature, probs=None, logits=None, validate_args=None):
+        self.temperature = temperature
+        if (probs is None) == (logits is None):
+            raise ValueError(
+                "Either `probs` or `logits` must be specified, but not both."
+            )
+        if probs is not None:
+            is_scalar = isinstance(probs, Number)
+            (self.probs,) = broadcast_all(probs)
+        else:
+            is_scalar = isinstance(logits, Number)
+            (self.logits,) = broadcast_all(logits)
+        self._param = self.probs if probs is not None else self.logits
+        if is_scalar:
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self._param.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(LogitRelaxedBernoulli, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.temperature = self.temperature
+        if "probs" in self.__dict__:
+            new.probs = self.probs.expand(batch_shape)
+            new._param = new.probs
+        if "logits" in self.__dict__:
+            new.logits = self.logits.expand(batch_shape)
+            new._param = new.logits
+        super(LogitRelaxedBernoulli, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._param.new(*args, **kwargs)
+
+    @lazy_property
+    def logits(self):
+        return probs_to_logits(self.probs, is_binary=True)
+
+    @lazy_property
+    def probs(self):
+        return logits_to_probs(self.logits, is_binary=True)
+
+    @property
+    def param_shape(self):
+        return self._param.size()
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        probs = clamp_probs(self.probs.expand(shape))
+        uniforms = clamp_probs(
+            torch.rand(shape, dtype=probs.dtype, device=probs.device)
+        )
+        return (
+            uniforms.log() - (-uniforms).log1p() + probs.log() - (-probs).log1p()
+        ) / self.temperature
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        logits, value = broadcast_all(self.logits, value)
+        diff = logits - value.mul(self.temperature)
+        return self.temperature.log() + diff - 2 * diff.exp().log1p()
+
+
+class RelaxedBernoulli(TransformedDistribution):
+    r"""
+    Creates a RelaxedBernoulli distribution, parametrized by
+    :attr:`temperature`, and either :attr:`probs` or :attr:`logits`
+    (but not both). This is a relaxed version of the `Bernoulli` distribution,
+    so the values are in (0, 1), and has reparametrizable samples.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = RelaxedBernoulli(torch.tensor([2.2]),
+        ...                      torch.tensor([0.1, 0.2, 0.3, 0.99]))
+        >>> m.sample()
+        tensor([ 0.2951,  0.3442,  0.8918,  0.9021])
+
+    Args:
+        temperature (Tensor): relaxation temperature
+        probs (Number, Tensor): the probability of sampling `1`
+        logits (Number, Tensor): the log-odds of sampling `1`
+    """
+    arg_constraints = {"probs": constraints.unit_interval, "logits": constraints.real}
+    support = constraints.unit_interval
+    has_rsample = True
+
+    def __init__(self, temperature, probs=None, logits=None, validate_args=None):
+        base_dist = LogitRelaxedBernoulli(temperature, probs, logits)
+        super().__init__(base_dist, SigmoidTransform(), validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(RelaxedBernoulli, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def temperature(self):
+        return self.base_dist.temperature
+
+    @property
+    def logits(self):
+        return self.base_dist.logits
+
+    @property
+    def probs(self):
+        return self.base_dist.probs

rl/Lib/site-packages/torch/distributions/relaxed_categorical.py

@@ -0,0 +1,142 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.distributions import constraints
+from torch.distributions.categorical import Categorical
+from torch.distributions.distribution import Distribution
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import ExpTransform
+from torch.distributions.utils import broadcast_all, clamp_probs
+from torch.types import _size
+
+
+__all__ = ["ExpRelaxedCategorical", "RelaxedOneHotCategorical"]
+
+
+class ExpRelaxedCategorical(Distribution):
+    r"""
+    Creates a ExpRelaxedCategorical parameterized by
+    :attr:`temperature`, and either :attr:`probs` or :attr:`logits` (but not both).
+    Returns the log of a point in the simplex. Based on the interface to
+    :class:`OneHotCategorical`.
+
+    Implementation based on [1].
+
+    See also: :func:`torch.distributions.OneHotCategorical`
+
+    Args:
+        temperature (Tensor): relaxation temperature
+        probs (Tensor): event probabilities
+        logits (Tensor): unnormalized log probability for each event
+
+    [1] The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables
+    (Maddison et al., 2017)
+
+    [2] Categorical Reparametrization with Gumbel-Softmax
+    (Jang et al., 2017)
+    """
+    arg_constraints = {"probs": constraints.simplex, "logits": constraints.real_vector}
+    support = (
+        constraints.real_vector
+    )  # The true support is actually a submanifold of this.
+    has_rsample = True
+
+    def __init__(self, temperature, probs=None, logits=None, validate_args=None):
+        self._categorical = Categorical(probs, logits)
+        self.temperature = temperature
+        batch_shape = self._categorical.batch_shape
+        event_shape = self._categorical.param_shape[-1:]
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(ExpRelaxedCategorical, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.temperature = self.temperature
+        new._categorical = self._categorical.expand(batch_shape)
+        super(ExpRelaxedCategorical, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    def _new(self, *args, **kwargs):
+        return self._categorical._new(*args, **kwargs)
+
+    @property
+    def param_shape(self):
+        return self._categorical.param_shape
+
+    @property
+    def logits(self):
+        return self._categorical.logits
+
+    @property
+    def probs(self):
+        return self._categorical.probs
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        uniforms = clamp_probs(
+            torch.rand(shape, dtype=self.logits.dtype, device=self.logits.device)
+        )
+        gumbels = -((-(uniforms.log())).log())
+        scores = (self.logits + gumbels) / self.temperature
+        return scores - scores.logsumexp(dim=-1, keepdim=True)
+
+    def log_prob(self, value):
+        K = self._categorical._num_events
+        if self._validate_args:
+            self._validate_sample(value)
+        logits, value = broadcast_all(self.logits, value)
+        log_scale = torch.full_like(
+            self.temperature, float(K)
+        ).lgamma() - self.temperature.log().mul(-(K - 1))
+        score = logits - value.mul(self.temperature)
+        score = (score - score.logsumexp(dim=-1, keepdim=True)).sum(-1)
+        return score + log_scale
+
+
+class RelaxedOneHotCategorical(TransformedDistribution):
+    r"""
+    Creates a RelaxedOneHotCategorical distribution parametrized by
+    :attr:`temperature`, and either :attr:`probs` or :attr:`logits`.
+    This is a relaxed version of the :class:`OneHotCategorical` distribution, so
+    its samples are on simplex, and are reparametrizable.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = RelaxedOneHotCategorical(torch.tensor([2.2]),
+        ...                              torch.tensor([0.1, 0.2, 0.3, 0.4]))
+        >>> m.sample()
+        tensor([ 0.1294,  0.2324,  0.3859,  0.2523])
+
+    Args:
+        temperature (Tensor): relaxation temperature
+        probs (Tensor): event probabilities
+        logits (Tensor): unnormalized log probability for each event
+    """
+    arg_constraints = {"probs": constraints.simplex, "logits": constraints.real_vector}
+    support = constraints.simplex
+    has_rsample = True
+
+    def __init__(self, temperature, probs=None, logits=None, validate_args=None):
+        base_dist = ExpRelaxedCategorical(
+            temperature, probs, logits, validate_args=validate_args
+        )
+        super().__init__(base_dist, ExpTransform(), validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(RelaxedOneHotCategorical, _instance)
+        return super().expand(batch_shape, _instance=new)
+
+    @property
+    def temperature(self):
+        return self.base_dist.temperature
+
+    @property
+    def logits(self):
+        return self.base_dist.logits
+
+    @property
+    def probs(self):
+        return self.base_dist.probs

rl/Lib/site-packages/torch/distributions/studentT.py

@@ -0,0 +1,119 @@
+# mypy: allow-untyped-defs
+import math
+
+import torch
+from torch import inf, nan
+from torch.distributions import Chi2, constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import _standard_normal, broadcast_all
+from torch.types import _size
+
+
+__all__ = ["StudentT"]
+
+
+class StudentT(Distribution):
+    r"""
+    Creates a Student's t-distribution parameterized by degree of
+    freedom :attr:`df`, mean :attr:`loc` and scale :attr:`scale`.
+
+    Example::
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = StudentT(torch.tensor([2.0]))
+        >>> m.sample()  # Student's t-distributed with degrees of freedom=2
+        tensor([ 0.1046])
+
+    Args:
+        df (float or Tensor): degrees of freedom
+        loc (float or Tensor): mean of the distribution
+        scale (float or Tensor): scale of the distribution
+    """
+    arg_constraints = {
+        "df": constraints.positive,
+        "loc": constraints.real,
+        "scale": constraints.positive,
+    }
+    support = constraints.real
+    has_rsample = True
+
+    @property
+    def mean(self):
+        m = self.loc.clone(memory_format=torch.contiguous_format)
+        m[self.df <= 1] = nan
+        return m
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @property
+    def variance(self):
+        m = self.df.clone(memory_format=torch.contiguous_format)
+        m[self.df > 2] = (
+            self.scale[self.df > 2].pow(2)
+            * self.df[self.df > 2]
+            / (self.df[self.df > 2] - 2)
+        )
+        m[(self.df <= 2) & (self.df > 1)] = inf
+        m[self.df <= 1] = nan
+        return m
+
+    def __init__(self, df, loc=0.0, scale=1.0, validate_args=None):
+        self.df, self.loc, self.scale = broadcast_all(df, loc, scale)
+        self._chi2 = Chi2(self.df)
+        batch_shape = self.df.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(StudentT, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.df = self.df.expand(batch_shape)
+        new.loc = self.loc.expand(batch_shape)
+        new.scale = self.scale.expand(batch_shape)
+        new._chi2 = self._chi2.expand(batch_shape)
+        super(StudentT, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        # NOTE: This does not agree with scipy implementation as much as other distributions.
+        # (see https://github.com/fritzo/notebooks/blob/master/debug-student-t.ipynb). Using DoubleTensor
+        # parameters seems to help.
+
+        #   X ~ Normal(0, 1)
+        #   Z ~ Chi2(df)
+        #   Y = X / sqrt(Z / df) ~ StudentT(df)
+        shape = self._extended_shape(sample_shape)
+        X = _standard_normal(shape, dtype=self.df.dtype, device=self.df.device)
+        Z = self._chi2.rsample(sample_shape)
+        Y = X * torch.rsqrt(Z / self.df)
+        return self.loc + self.scale * Y
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        y = (value - self.loc) / self.scale
+        Z = (
+            self.scale.log()
+            + 0.5 * self.df.log()
+            + 0.5 * math.log(math.pi)
+            + torch.lgamma(0.5 * self.df)
+            - torch.lgamma(0.5 * (self.df + 1.0))
+        )
+        return -0.5 * (self.df + 1.0) * torch.log1p(y**2.0 / self.df) - Z
+
+    def entropy(self):
+        lbeta = (
+            torch.lgamma(0.5 * self.df)
+            + math.lgamma(0.5)
+            - torch.lgamma(0.5 * (self.df + 1))
+        )
+        return (
+            self.scale.log()
+            + 0.5
+            * (self.df + 1)
+            * (torch.digamma(0.5 * (self.df + 1)) - torch.digamma(0.5 * self.df))
+            + 0.5 * self.df.log()
+            + lbeta
+        )

rl/Lib/site-packages/torch/distributions/transformed_distribution.py

@@ -0,0 +1,216 @@
+# mypy: allow-untyped-defs
+from typing import Dict
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.independent import Independent
+from torch.distributions.transforms import ComposeTransform, Transform
+from torch.distributions.utils import _sum_rightmost
+from torch.types import _size
+
+
+__all__ = ["TransformedDistribution"]
+
+
+class TransformedDistribution(Distribution):
+    r"""
+    Extension of the Distribution class, which applies a sequence of Transforms
+    to a base distribution.  Let f be the composition of transforms applied::
+
+        X ~ BaseDistribution
+        Y = f(X) ~ TransformedDistribution(BaseDistribution, f)
+        log p(Y) = log p(X) + log |det (dX/dY)|
+
+    Note that the ``.event_shape`` of a :class:`TransformedDistribution` is the
+    maximum shape of its base distribution and its transforms, since transforms
+    can introduce correlations among events.
+
+    An example for the usage of :class:`TransformedDistribution` would be::
+
+        # Building a Logistic Distribution
+        # X ~ Uniform(0, 1)
+        # f = a + b * logit(X)
+        # Y ~ f(X) ~ Logistic(a, b)
+        base_distribution = Uniform(0, 1)
+        transforms = [SigmoidTransform().inv, AffineTransform(loc=a, scale=b)]
+        logistic = TransformedDistribution(base_distribution, transforms)
+
+    For more examples, please look at the implementations of
+    :class:`~torch.distributions.gumbel.Gumbel`,
+    :class:`~torch.distributions.half_cauchy.HalfCauchy`,
+    :class:`~torch.distributions.half_normal.HalfNormal`,
+    :class:`~torch.distributions.log_normal.LogNormal`,
+    :class:`~torch.distributions.pareto.Pareto`,
+    :class:`~torch.distributions.weibull.Weibull`,
+    :class:`~torch.distributions.relaxed_bernoulli.RelaxedBernoulli` and
+    :class:`~torch.distributions.relaxed_categorical.RelaxedOneHotCategorical`
+    """
+    arg_constraints: Dict[str, constraints.Constraint] = {}
+
+    def __init__(self, base_distribution, transforms, validate_args=None):
+        if isinstance(transforms, Transform):
+            self.transforms = [
+                transforms,
+            ]
+        elif isinstance(transforms, list):
+            if not all(isinstance(t, Transform) for t in transforms):
+                raise ValueError(
+                    "transforms must be a Transform or a list of Transforms"
+                )
+            self.transforms = transforms
+        else:
+            raise ValueError(
+                f"transforms must be a Transform or list, but was {transforms}"
+            )
+
+        # Reshape base_distribution according to transforms.
+        base_shape = base_distribution.batch_shape + base_distribution.event_shape
+        base_event_dim = len(base_distribution.event_shape)
+        transform = ComposeTransform(self.transforms)
+        if len(base_shape) < transform.domain.event_dim:
+            raise ValueError(
+                f"base_distribution needs to have shape with size at least {transform.domain.event_dim}, but got {base_shape}."
+            )
+        forward_shape = transform.forward_shape(base_shape)
+        expanded_base_shape = transform.inverse_shape(forward_shape)
+        if base_shape != expanded_base_shape:
+            base_batch_shape = expanded_base_shape[
+                : len(expanded_base_shape) - base_event_dim
+            ]
+            base_distribution = base_distribution.expand(base_batch_shape)
+        reinterpreted_batch_ndims = transform.domain.event_dim - base_event_dim
+        if reinterpreted_batch_ndims > 0:
+            base_distribution = Independent(
+                base_distribution, reinterpreted_batch_ndims
+            )
+        self.base_dist = base_distribution
+
+        # Compute shapes.
+        transform_change_in_event_dim = (
+            transform.codomain.event_dim - transform.domain.event_dim
+        )
+        event_dim = max(
+            transform.codomain.event_dim,  # the transform is coupled
+            base_event_dim + transform_change_in_event_dim,  # the base dist is coupled
+        )
+        assert len(forward_shape) >= event_dim
+        cut = len(forward_shape) - event_dim
+        batch_shape = forward_shape[:cut]
+        event_shape = forward_shape[cut:]
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(TransformedDistribution, _instance)
+        batch_shape = torch.Size(batch_shape)
+        shape = batch_shape + self.event_shape
+        for t in reversed(self.transforms):
+            shape = t.inverse_shape(shape)
+        base_batch_shape = shape[: len(shape) - len(self.base_dist.event_shape)]
+        new.base_dist = self.base_dist.expand(base_batch_shape)
+        new.transforms = self.transforms
+        super(TransformedDistribution, new).__init__(
+            batch_shape, self.event_shape, validate_args=False
+        )
+        new._validate_args = self._validate_args
+        return new
+
+    @constraints.dependent_property(is_discrete=False)
+    def support(self):
+        if not self.transforms:
+            return self.base_dist.support
+        support = self.transforms[-1].codomain
+        if len(self.event_shape) > support.event_dim:
+            support = constraints.independent(
+                support, len(self.event_shape) - support.event_dim
+            )
+        return support
+
+    @property
+    def has_rsample(self):
+        return self.base_dist.has_rsample
+
+    def sample(self, sample_shape=torch.Size()):
+        """
+        Generates a sample_shape shaped sample or sample_shape shaped batch of
+        samples if the distribution parameters are batched. Samples first from
+        base distribution and applies `transform()` for every transform in the
+        list.
+        """
+        with torch.no_grad():
+            x = self.base_dist.sample(sample_shape)
+            for transform in self.transforms:
+                x = transform(x)
+            return x
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        """
+        Generates a sample_shape shaped reparameterized sample or sample_shape
+        shaped batch of reparameterized samples if the distribution parameters
+        are batched. Samples first from base distribution and applies
+        `transform()` for every transform in the list.
+        """
+        x = self.base_dist.rsample(sample_shape)
+        for transform in self.transforms:
+            x = transform(x)
+        return x
+
+    def log_prob(self, value):
+        """
+        Scores the sample by inverting the transform(s) and computing the score
+        using the score of the base distribution and the log abs det jacobian.
+        """
+        if self._validate_args:
+            self._validate_sample(value)
+        event_dim = len(self.event_shape)
+        log_prob = 0.0
+        y = value
+        for transform in reversed(self.transforms):
+            x = transform.inv(y)
+            event_dim += transform.domain.event_dim - transform.codomain.event_dim
+            log_prob = log_prob - _sum_rightmost(
+                transform.log_abs_det_jacobian(x, y),
+                event_dim - transform.domain.event_dim,
+            )
+            y = x
+
+        log_prob = log_prob + _sum_rightmost(
+            self.base_dist.log_prob(y), event_dim - len(self.base_dist.event_shape)
+        )
+        return log_prob
+
+    def _monotonize_cdf(self, value):
+        """
+        This conditionally flips ``value -> 1-value`` to ensure :meth:`cdf` is
+        monotone increasing.
+        """
+        sign = 1
+        for transform in self.transforms:
+            sign = sign * transform.sign
+        if isinstance(sign, int) and sign == 1:
+            return value
+        return sign * (value - 0.5) + 0.5
+
+    def cdf(self, value):
+        """
+        Computes the cumulative distribution function by inverting the
+        transform(s) and computing the score of the base distribution.
+        """
+        for transform in self.transforms[::-1]:
+            value = transform.inv(value)
+        if self._validate_args:
+            self.base_dist._validate_sample(value)
+        value = self.base_dist.cdf(value)
+        value = self._monotonize_cdf(value)
+        return value
+
+    def icdf(self, value):
+        """
+        Computes the inverse cumulative distribution function using
+        transform(s) and computing the score of the base distribution.
+        """
+        value = self._monotonize_cdf(value)
+        value = self.base_dist.icdf(value)
+        for transform in self.transforms:
+            value = transform(value)
+        return value

rl/Lib/site-packages/torch/distributions/uniform.py

@@ -0,0 +1,102 @@
+# mypy: allow-untyped-defs
+from numbers import Number
+
+import torch
+from torch import nan
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import broadcast_all
+from torch.types import _size
+
+
+__all__ = ["Uniform"]
+
+
+class Uniform(Distribution):
+    r"""
+    Generates uniformly distributed random samples from the half-open interval
+    ``[low, high)``.
+
+    Example::
+
+        >>> m = Uniform(torch.tensor([0.0]), torch.tensor([5.0]))
+        >>> m.sample()  # uniformly distributed in the range [0.0, 5.0)
+        >>> # xdoctest: +SKIP
+        tensor([ 2.3418])
+
+    Args:
+        low (float or Tensor): lower range (inclusive).
+        high (float or Tensor): upper range (exclusive).
+    """
+    # TODO allow (loc,scale) parameterization to allow independent constraints.
+    arg_constraints = {
+        "low": constraints.dependent(is_discrete=False, event_dim=0),
+        "high": constraints.dependent(is_discrete=False, event_dim=0),
+    }
+    has_rsample = True
+
+    @property
+    def mean(self):
+        return (self.high + self.low) / 2
+
+    @property
+    def mode(self):
+        return nan * self.high
+
+    @property
+    def stddev(self):
+        return (self.high - self.low) / 12**0.5
+
+    @property
+    def variance(self):
+        return (self.high - self.low).pow(2) / 12
+
+    def __init__(self, low, high, validate_args=None):
+        self.low, self.high = broadcast_all(low, high)
+
+        if isinstance(low, Number) and isinstance(high, Number):
+            batch_shape = torch.Size()
+        else:
+            batch_shape = self.low.size()
+        super().__init__(batch_shape, validate_args=validate_args)
+
+        if self._validate_args and not torch.lt(self.low, self.high).all():
+            raise ValueError("Uniform is not defined when low>= high")
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Uniform, _instance)
+        batch_shape = torch.Size(batch_shape)
+        new.low = self.low.expand(batch_shape)
+        new.high = self.high.expand(batch_shape)
+        super(Uniform, new).__init__(batch_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @constraints.dependent_property(is_discrete=False, event_dim=0)
+    def support(self):
+        return constraints.interval(self.low, self.high)
+
+    def rsample(self, sample_shape: _size = torch.Size()) -> torch.Tensor:
+        shape = self._extended_shape(sample_shape)
+        rand = torch.rand(shape, dtype=self.low.dtype, device=self.low.device)
+        return self.low + rand * (self.high - self.low)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        lb = self.low.le(value).type_as(self.low)
+        ub = self.high.gt(value).type_as(self.low)
+        return torch.log(lb.mul(ub)) - torch.log(self.high - self.low)
+
+    def cdf(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        result = (value - self.low) / (self.high - self.low)
+        return result.clamp(min=0, max=1)
+
+    def icdf(self, value):
+        result = value * (self.high - self.low) + self.low
+        return result
+
+    def entropy(self):
+        return torch.log(self.high - self.low)

rl/Lib/site-packages/torch/distributions/utils.py

@@ -0,0 +1,200 @@
+# mypy: allow-untyped-defs
+from functools import update_wrapper
+from numbers import Number
+from typing import Any, Dict
+
+import torch
+import torch.nn.functional as F
+from torch.overrides import is_tensor_like
+
+
+euler_constant = 0.57721566490153286060  # Euler Mascheroni Constant
+
+__all__ = [
+    "broadcast_all",
+    "logits_to_probs",
+    "clamp_probs",
+    "probs_to_logits",
+    "lazy_property",
+    "tril_matrix_to_vec",
+    "vec_to_tril_matrix",
+]
+
+
+def broadcast_all(*values):
+    r"""
+    Given a list of values (possibly containing numbers), returns a list where each
+    value is broadcasted based on the following rules:
+      - `torch.*Tensor` instances are broadcasted as per :ref:`_broadcasting-semantics`.
+      - numbers.Number instances (scalars) are upcast to tensors having
+        the same size and type as the first tensor passed to `values`.  If all the
+        values are scalars, then they are upcasted to scalar Tensors.
+
+    Args:
+        values (list of `numbers.Number`, `torch.*Tensor` or objects implementing __torch_function__)
+
+    Raises:
+        ValueError: if any of the values is not a `numbers.Number` instance,
+            a `torch.*Tensor` instance, or an instance implementing __torch_function__
+    """
+    if not all(is_tensor_like(v) or isinstance(v, Number) for v in values):
+        raise ValueError(
+            "Input arguments must all be instances of numbers.Number, "
+            "torch.Tensor or objects implementing __torch_function__."
+        )
+    if not all(is_tensor_like(v) for v in values):
+        options: Dict[str, Any] = dict(dtype=torch.get_default_dtype())
+        for value in values:
+            if isinstance(value, torch.Tensor):
+                options = dict(dtype=value.dtype, device=value.device)
+                break
+        new_values = [
+            v if is_tensor_like(v) else torch.tensor(v, **options) for v in values
+        ]
+        return torch.broadcast_tensors(*new_values)
+    return torch.broadcast_tensors(*values)
+
+
+def _standard_normal(shape, dtype, device):
+    if torch._C._get_tracing_state():
+        # [JIT WORKAROUND] lack of support for .normal_()
+        return torch.normal(
+            torch.zeros(shape, dtype=dtype, device=device),
+            torch.ones(shape, dtype=dtype, device=device),
+        )
+    return torch.empty(shape, dtype=dtype, device=device).normal_()
+
+
+def _sum_rightmost(value, dim):
+    r"""
+    Sum out ``dim`` many rightmost dimensions of a given tensor.
+
+    Args:
+        value (Tensor): A tensor of ``.dim()`` at least ``dim``.
+        dim (int): The number of rightmost dims to sum out.
+    """
+    if dim == 0:
+        return value
+    required_shape = value.shape[:-dim] + (-1,)
+    return value.reshape(required_shape).sum(-1)
+
+
+def logits_to_probs(logits, is_binary=False):
+    r"""
+    Converts a tensor of logits into probabilities. Note that for the
+    binary case, each value denotes log odds, whereas for the
+    multi-dimensional case, the values along the last dimension denote
+    the log probabilities (possibly unnormalized) of the events.
+    """
+    if is_binary:
+        return torch.sigmoid(logits)
+    return F.softmax(logits, dim=-1)
+
+
+def clamp_probs(probs):
+    """Clamps the probabilities to be in the open interval `(0, 1)`.
+
+    The probabilities would be clamped between `eps` and `1 - eps`,
+    and `eps` would be the smallest representable positive number for the input data type.
+
+    Args:
+        probs (Tensor): A tensor of probabilities.
+
+    Returns:
+        Tensor: The clamped probabilities.
+
+    Examples:
+        >>> probs = torch.tensor([0.0, 0.5, 1.0])
+        >>> clamp_probs(probs)
+        tensor([1.1921e-07, 5.0000e-01, 1.0000e+00])
+
+        >>> probs = torch.tensor([0.0, 0.5, 1.0], dtype=torch.float64)
+        >>> clamp_probs(probs)
+        tensor([2.2204e-16, 5.0000e-01, 1.0000e+00], dtype=torch.float64)
+
+    """
+    eps = torch.finfo(probs.dtype).eps
+    return probs.clamp(min=eps, max=1 - eps)
+
+
+def probs_to_logits(probs, is_binary=False):
+    r"""
+    Converts a tensor of probabilities into logits. For the binary case,
+    this denotes the probability of occurrence of the event indexed by `1`.
+    For the multi-dimensional case, the values along the last dimension
+    denote the probabilities of occurrence of each of the events.
+    """
+    ps_clamped = clamp_probs(probs)
+    if is_binary:
+        return torch.log(ps_clamped) - torch.log1p(-ps_clamped)
+    return torch.log(ps_clamped)
+
+
+class lazy_property:
+    r"""
+    Used as a decorator for lazy loading of class attributes. This uses a
+    non-data descriptor that calls the wrapped method to compute the property on
+    first call; thereafter replacing the wrapped method into an instance
+    attribute.
+    """
+
+    def __init__(self, wrapped):
+        self.wrapped = wrapped
+        update_wrapper(self, wrapped)  # type:ignore[arg-type]
+
+    def __get__(self, instance, obj_type=None):
+        if instance is None:
+            return _lazy_property_and_property(self.wrapped)
+        with torch.enable_grad():
+            value = self.wrapped(instance)
+        setattr(instance, self.wrapped.__name__, value)
+        return value
+
+
+class _lazy_property_and_property(lazy_property, property):
+    """We want lazy properties to look like multiple things.
+
+    * property when Sphinx autodoc looks
+    * lazy_property when Distribution validate_args looks
+    """
+
+    def __init__(self, wrapped):
+        property.__init__(self, wrapped)
+
+
+def tril_matrix_to_vec(mat: torch.Tensor, diag: int = 0) -> torch.Tensor:
+    r"""
+    Convert a `D x D` matrix or a batch of matrices into a (batched) vector
+    which comprises of lower triangular elements from the matrix in row order.
+    """
+    n = mat.shape[-1]
+    if not torch._C._get_tracing_state() and (diag < -n or diag >= n):
+        raise ValueError(f"diag ({diag}) provided is outside [{-n}, {n-1}].")
+    arange = torch.arange(n, device=mat.device)
+    tril_mask = arange < arange.view(-1, 1) + (diag + 1)
+    vec = mat[..., tril_mask]
+    return vec
+
+
+def vec_to_tril_matrix(vec: torch.Tensor, diag: int = 0) -> torch.Tensor:
+    r"""
+    Convert a vector or a batch of vectors into a batched `D x D`
+    lower triangular matrix containing elements from the vector in row order.
+    """
+    # +ve root of D**2 + (1+2*diag)*D - |diag| * (diag+1) - 2*vec.shape[-1] = 0
+    n = (
+        -(1 + 2 * diag)
+        + ((1 + 2 * diag) ** 2 + 8 * vec.shape[-1] + 4 * abs(diag) * (diag + 1)) ** 0.5
+    ) / 2
+    eps = torch.finfo(vec.dtype).eps
+    if not torch._C._get_tracing_state() and (round(n) - n > eps):
+        raise ValueError(
+            f"The size of last dimension is {vec.shape[-1]} which cannot be expressed as "
+            + "the lower triangular part of a square D x D matrix."
+        )
+    n = round(n.item()) if isinstance(n, torch.Tensor) else round(n)
+    mat = vec.new_zeros(vec.shape[:-1] + torch.Size((n, n)))
+    arange = torch.arange(n, device=vec.device)
+    tril_mask = arange < arange.view(-1, 1) + (diag + 1)
+    mat[..., tril_mask] = vec
+    return mat

rl/Lib/site-packages/torch/distributions/von_mises.py

@@ -0,0 +1,211 @@
+# mypy: allow-untyped-defs
+import math
+
+import torch
+import torch.jit
+from torch.distributions import constraints
+from torch.distributions.distribution import Distribution
+from torch.distributions.utils import broadcast_all, lazy_property
+
+
+__all__ = ["VonMises"]
+
+
+def _eval_poly(y, coef):
+    coef = list(coef)
+    result = coef.pop()
+    while coef:
+        result = coef.pop() + y * result
+    return result
+
+
+_I0_COEF_SMALL = [
+    1.0,
+    3.5156229,
+    3.0899424,
+    1.2067492,
+    0.2659732,
+    0.360768e-1,
+    0.45813e-2,
+]
+_I0_COEF_LARGE = [
+    0.39894228,
+    0.1328592e-1,
+    0.225319e-2,
+    -0.157565e-2,
+    0.916281e-2,
+    -0.2057706e-1,
+    0.2635537e-1,
+    -0.1647633e-1,
+    0.392377e-2,
+]
+_I1_COEF_SMALL = [
+    0.5,
+    0.87890594,
+    0.51498869,
+    0.15084934,
+    0.2658733e-1,
+    0.301532e-2,
+    0.32411e-3,
+]
+_I1_COEF_LARGE = [
+    0.39894228,
+    -0.3988024e-1,
+    -0.362018e-2,
+    0.163801e-2,
+    -0.1031555e-1,
+    0.2282967e-1,
+    -0.2895312e-1,
+    0.1787654e-1,
+    -0.420059e-2,
+]
+
+_COEF_SMALL = [_I0_COEF_SMALL, _I1_COEF_SMALL]
+_COEF_LARGE = [_I0_COEF_LARGE, _I1_COEF_LARGE]
+
+
+def _log_modified_bessel_fn(x, order=0):
+    """
+    Returns ``log(I_order(x))`` for ``x > 0``,
+    where `order` is either 0 or 1.
+    """
+    assert order == 0 or order == 1
+
+    # compute small solution
+    y = x / 3.75
+    y = y * y
+    small = _eval_poly(y, _COEF_SMALL[order])
+    if order == 1:
+        small = x.abs() * small
+    small = small.log()
+
+    # compute large solution
+    y = 3.75 / x
+    large = x - 0.5 * x.log() + _eval_poly(y, _COEF_LARGE[order]).log()
+
+    result = torch.where(x < 3.75, small, large)
+    return result
+
+
+@torch.jit.script_if_tracing
+def _rejection_sample(loc, concentration, proposal_r, x):
+    done = torch.zeros(x.shape, dtype=torch.bool, device=loc.device)
+    while not done.all():
+        u = torch.rand((3,) + x.shape, dtype=loc.dtype, device=loc.device)
+        u1, u2, u3 = u.unbind()
+        z = torch.cos(math.pi * u1)
+        f = (1 + proposal_r * z) / (proposal_r + z)
+        c = concentration * (proposal_r - f)
+        accept = ((c * (2 - c) - u2) > 0) | ((c / u2).log() + 1 - c >= 0)
+        if accept.any():
+            x = torch.where(accept, (u3 - 0.5).sign() * f.acos(), x)
+            done = done | accept
+    return (x + math.pi + loc) % (2 * math.pi) - math.pi
+
+
+class VonMises(Distribution):
+    """
+    A circular von Mises distribution.
+
+    This implementation uses polar coordinates. The ``loc`` and ``value`` args
+    can be any real number (to facilitate unconstrained optimization), but are
+    interpreted as angles modulo 2 pi.
+
+    Example::
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = VonMises(torch.tensor([1.0]), torch.tensor([1.0]))
+        >>> m.sample()  # von Mises distributed with loc=1 and concentration=1
+        tensor([1.9777])
+
+    :param torch.Tensor loc: an angle in radians.
+    :param torch.Tensor concentration: concentration parameter
+    """
+
+    arg_constraints = {"loc": constraints.real, "concentration": constraints.positive}
+    support = constraints.real
+    has_rsample = False
+
+    def __init__(self, loc, concentration, validate_args=None):
+        self.loc, self.concentration = broadcast_all(loc, concentration)
+        batch_shape = self.loc.shape
+        event_shape = torch.Size()
+        super().__init__(batch_shape, event_shape, validate_args)
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        log_prob = self.concentration * torch.cos(value - self.loc)
+        log_prob = (
+            log_prob
+            - math.log(2 * math.pi)
+            - _log_modified_bessel_fn(self.concentration, order=0)
+        )
+        return log_prob
+
+    @lazy_property
+    def _loc(self):
+        return self.loc.to(torch.double)
+
+    @lazy_property
+    def _concentration(self):
+        return self.concentration.to(torch.double)
+
+    @lazy_property
+    def _proposal_r(self):
+        kappa = self._concentration
+        tau = 1 + (1 + 4 * kappa**2).sqrt()
+        rho = (tau - (2 * tau).sqrt()) / (2 * kappa)
+        _proposal_r = (1 + rho**2) / (2 * rho)
+        # second order Taylor expansion around 0 for small kappa
+        _proposal_r_taylor = 1 / kappa + kappa
+        return torch.where(kappa < 1e-5, _proposal_r_taylor, _proposal_r)
+
+    @torch.no_grad()
+    def sample(self, sample_shape=torch.Size()):
+        """
+        The sampling algorithm for the von Mises distribution is based on the
+        following paper: D.J. Best and N.I. Fisher, "Efficient simulation of the
+        von Mises distribution." Applied Statistics (1979): 152-157.
+
+        Sampling is always done in double precision internally to avoid a hang
+        in _rejection_sample() for small values of the concentration, which
+        starts to happen for single precision around 1e-4 (see issue #88443).
+        """
+        shape = self._extended_shape(sample_shape)
+        x = torch.empty(shape, dtype=self._loc.dtype, device=self.loc.device)
+        return _rejection_sample(
+            self._loc, self._concentration, self._proposal_r, x
+        ).to(self.loc.dtype)
+
+    def expand(self, batch_shape):
+        try:
+            return super().expand(batch_shape)
+        except NotImplementedError:
+            validate_args = self.__dict__.get("_validate_args")
+            loc = self.loc.expand(batch_shape)
+            concentration = self.concentration.expand(batch_shape)
+            return type(self)(loc, concentration, validate_args=validate_args)
+
+    @property
+    def mean(self):
+        """
+        The provided mean is the circular one.
+        """
+        return self.loc
+
+    @property
+    def mode(self):
+        return self.loc
+
+    @lazy_property
+    def variance(self):
+        """
+        The provided variance is the circular one.
+        """
+        return (
+            1
+            - (
+                _log_modified_bessel_fn(self.concentration, order=1)
+                - _log_modified_bessel_fn(self.concentration, order=0)
+            ).exp()
+        )

rl/Lib/site-packages/torch/distributions/weibull.py

@@ -0,0 +1,85 @@
+# mypy: allow-untyped-defs
+import torch
+from torch.distributions import constraints
+from torch.distributions.exponential import Exponential
+from torch.distributions.gumbel import euler_constant
+from torch.distributions.transformed_distribution import TransformedDistribution
+from torch.distributions.transforms import AffineTransform, PowerTransform
+from torch.distributions.utils import broadcast_all
+
+
+__all__ = ["Weibull"]
+
+
+class Weibull(TransformedDistribution):
+    r"""
+    Samples from a two-parameter Weibull distribution.
+
+    Example:
+
+        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
+        >>> m = Weibull(torch.tensor([1.0]), torch.tensor([1.0]))
+        >>> m.sample()  # sample from a Weibull distribution with scale=1, concentration=1
+        tensor([ 0.4784])
+
+    Args:
+        scale (float or Tensor): Scale parameter of distribution (lambda).
+        concentration (float or Tensor): Concentration parameter of distribution (k/shape).
+    """
+    arg_constraints = {
+        "scale": constraints.positive,
+        "concentration": constraints.positive,
+    }
+    support = constraints.positive
+
+    def __init__(self, scale, concentration, validate_args=None):
+        self.scale, self.concentration = broadcast_all(scale, concentration)
+        self.concentration_reciprocal = self.concentration.reciprocal()
+        base_dist = Exponential(
+            torch.ones_like(self.scale), validate_args=validate_args
+        )
+        transforms = [
+            PowerTransform(exponent=self.concentration_reciprocal),
+            AffineTransform(loc=0, scale=self.scale),
+        ]
+        super().__init__(base_dist, transforms, validate_args=validate_args)
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Weibull, _instance)
+        new.scale = self.scale.expand(batch_shape)
+        new.concentration = self.concentration.expand(batch_shape)
+        new.concentration_reciprocal = new.concentration.reciprocal()
+        base_dist = self.base_dist.expand(batch_shape)
+        transforms = [
+            PowerTransform(exponent=new.concentration_reciprocal),
+            AffineTransform(loc=0, scale=new.scale),
+        ]
+        super(Weibull, new).__init__(base_dist, transforms, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @property
+    def mean(self):
+        return self.scale * torch.exp(torch.lgamma(1 + self.concentration_reciprocal))
+
+    @property
+    def mode(self):
+        return (
+            self.scale
+            * ((self.concentration - 1) / self.concentration)
+            ** self.concentration.reciprocal()
+        )
+
+    @property
+    def variance(self):
+        return self.scale.pow(2) * (
+            torch.exp(torch.lgamma(1 + 2 * self.concentration_reciprocal))
+            - torch.exp(2 * torch.lgamma(1 + self.concentration_reciprocal))
+        )
+
+    def entropy(self):
+        return (
+            euler_constant * (1 - self.concentration_reciprocal)
+            + torch.log(self.scale * self.concentration_reciprocal)
+            + 1
+        )

rl/Lib/site-packages/torch/distributions/wishart.py

@@ -0,0 +1,339 @@
+# mypy: allow-untyped-defs
+import math
+import warnings
+from numbers import Number
+from typing import Optional, Union
+
+import torch
+from torch import nan
+from torch.distributions import constraints
+from torch.distributions.exp_family import ExponentialFamily
+from torch.distributions.multivariate_normal import _precision_to_scale_tril
+from torch.distributions.utils import lazy_property
+from torch.types import _size
+
+
+__all__ = ["Wishart"]
+
+_log_2 = math.log(2)
+
+
+def _mvdigamma(x: torch.Tensor, p: int) -> torch.Tensor:
+    assert x.gt((p - 1) / 2).all(), "Wrong domain for multivariate digamma function."
+    return torch.digamma(
+        x.unsqueeze(-1)
+        - torch.arange(p, dtype=x.dtype, device=x.device).div(2).expand(x.shape + (-1,))
+    ).sum(-1)
+
+
+def _clamp_above_eps(x: torch.Tensor) -> torch.Tensor:
+    # We assume positive input for this function
+    return x.clamp(min=torch.finfo(x.dtype).eps)
+
+
+class Wishart(ExponentialFamily):
+    r"""
+    Creates a Wishart distribution parameterized by a symmetric positive definite matrix :math:`\Sigma`,
+    or its Cholesky decomposition :math:`\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top`
+
+    Example:
+        >>> # xdoctest: +SKIP("FIXME: scale_tril must be at least two-dimensional")
+        >>> m = Wishart(torch.Tensor([2]), covariance_matrix=torch.eye(2))
+        >>> m.sample()  # Wishart distributed with mean=`df * I` and
+        >>>             # variance(x_ij)=`df` for i != j and variance(x_ij)=`2 * df` for i == j
+
+    Args:
+        df (float or Tensor): real-valued parameter larger than the (dimension of Square matrix) - 1
+        covariance_matrix (Tensor): positive-definite covariance matrix
+        precision_matrix (Tensor): positive-definite precision matrix
+        scale_tril (Tensor): lower-triangular factor of covariance, with positive-valued diagonal
+    Note:
+        Only one of :attr:`covariance_matrix` or :attr:`precision_matrix` or
+        :attr:`scale_tril` can be specified.
+        Using :attr:`scale_tril` will be more efficient: all computations internally
+        are based on :attr:`scale_tril`. If :attr:`covariance_matrix` or
+        :attr:`precision_matrix` is passed instead, it is only used to compute
+        the corresponding lower triangular matrices using a Cholesky decomposition.
+        'torch.distributions.LKJCholesky' is a restricted Wishart distribution.[1]
+
+    **References**
+
+    [1] Wang, Z., Wu, Y. and Chu, H., 2018. `On equivalence of the LKJ distribution and the restricted Wishart distribution`.
+    [2] Sawyer, S., 2007. `Wishart Distributions and Inverse-Wishart Sampling`.
+    [3] Anderson, T. W., 2003. `An Introduction to Multivariate Statistical Analysis (3rd ed.)`.
+    [4] Odell, P. L. & Feiveson, A. H., 1966. `A Numerical Procedure to Generate a SampleCovariance Matrix`. JASA, 61(313):199-203.
+    [5] Ku, Y.-C. & Bloomfield, P., 2010. `Generating Random Wishart Matrices with Fractional Degrees of Freedom in OX`.
+    """
+    arg_constraints = {
+        "covariance_matrix": constraints.positive_definite,
+        "precision_matrix": constraints.positive_definite,
+        "scale_tril": constraints.lower_cholesky,
+        "df": constraints.greater_than(0),
+    }
+    support = constraints.positive_definite
+    has_rsample = True
+    _mean_carrier_measure = 0
+
+    def __init__(
+        self,
+        df: Union[torch.Tensor, Number],
+        covariance_matrix: Optional[torch.Tensor] = None,
+        precision_matrix: Optional[torch.Tensor] = None,
+        scale_tril: Optional[torch.Tensor] = None,
+        validate_args=None,
+    ):
+        assert (covariance_matrix is not None) + (scale_tril is not None) + (
+            precision_matrix is not None
+        ) == 1, "Exactly one of covariance_matrix or precision_matrix or scale_tril may be specified."
+
+        param = next(
+            p
+            for p in (covariance_matrix, precision_matrix, scale_tril)
+            if p is not None
+        )
+
+        if param.dim() < 2:
+            raise ValueError(
+                "scale_tril must be at least two-dimensional, with optional leading batch dimensions"
+            )
+
+        if isinstance(df, Number):
+            batch_shape = torch.Size(param.shape[:-2])
+            self.df = torch.tensor(df, dtype=param.dtype, device=param.device)
+        else:
+            batch_shape = torch.broadcast_shapes(param.shape[:-2], df.shape)
+            self.df = df.expand(batch_shape)
+        event_shape = param.shape[-2:]
+
+        if self.df.le(event_shape[-1] - 1).any():
+            raise ValueError(
+                f"Value of df={df} expected to be greater than ndim - 1 = {event_shape[-1]-1}."
+            )
+
+        if scale_tril is not None:
+            self.scale_tril = param.expand(batch_shape + (-1, -1))
+        elif covariance_matrix is not None:
+            self.covariance_matrix = param.expand(batch_shape + (-1, -1))
+        elif precision_matrix is not None:
+            self.precision_matrix = param.expand(batch_shape + (-1, -1))
+
+        self.arg_constraints["df"] = constraints.greater_than(event_shape[-1] - 1)
+        if self.df.lt(event_shape[-1]).any():
+            warnings.warn(
+                "Low df values detected. Singular samples are highly likely to occur for ndim - 1 < df < ndim."
+            )
+
+        super().__init__(batch_shape, event_shape, validate_args=validate_args)
+        self._batch_dims = [-(x + 1) for x in range(len(self._batch_shape))]
+
+        if scale_tril is not None:
+            self._unbroadcasted_scale_tril = scale_tril
+        elif covariance_matrix is not None:
+            self._unbroadcasted_scale_tril = torch.linalg.cholesky(covariance_matrix)
+        else:  # precision_matrix is not None
+            self._unbroadcasted_scale_tril = _precision_to_scale_tril(precision_matrix)
+
+        # Chi2 distribution is needed for Bartlett decomposition sampling
+        self._dist_chi2 = torch.distributions.chi2.Chi2(
+            df=(
+                self.df.unsqueeze(-1)
+                - torch.arange(
+                    self._event_shape[-1],
+                    dtype=self._unbroadcasted_scale_tril.dtype,
+                    device=self._unbroadcasted_scale_tril.device,
+                ).expand(batch_shape + (-1,))
+            )
+        )
+
+    def expand(self, batch_shape, _instance=None):
+        new = self._get_checked_instance(Wishart, _instance)
+        batch_shape = torch.Size(batch_shape)
+        cov_shape = batch_shape + self.event_shape
+        new._unbroadcasted_scale_tril = self._unbroadcasted_scale_tril.expand(cov_shape)
+        new.df = self.df.expand(batch_shape)
+
+        new._batch_dims = [-(x + 1) for x in range(len(batch_shape))]
+
+        if "covariance_matrix" in self.__dict__:
+            new.covariance_matrix = self.covariance_matrix.expand(cov_shape)
+        if "scale_tril" in self.__dict__:
+            new.scale_tril = self.scale_tril.expand(cov_shape)
+        if "precision_matrix" in self.__dict__:
+            new.precision_matrix = self.precision_matrix.expand(cov_shape)
+
+        # Chi2 distribution is needed for Bartlett decomposition sampling
+        new._dist_chi2 = torch.distributions.chi2.Chi2(
+            df=(
+                new.df.unsqueeze(-1)
+                - torch.arange(
+                    self.event_shape[-1],
+                    dtype=new._unbroadcasted_scale_tril.dtype,
+                    device=new._unbroadcasted_scale_tril.device,
+                ).expand(batch_shape + (-1,))
+            )
+        )
+
+        super(Wishart, new).__init__(batch_shape, self.event_shape, validate_args=False)
+        new._validate_args = self._validate_args
+        return new
+
+    @lazy_property
+    def scale_tril(self):
+        return self._unbroadcasted_scale_tril.expand(
+            self._batch_shape + self._event_shape
+        )
+
+    @lazy_property
+    def covariance_matrix(self):
+        return (
+            self._unbroadcasted_scale_tril
+            @ self._unbroadcasted_scale_tril.transpose(-2, -1)
+        ).expand(self._batch_shape + self._event_shape)
+
+    @lazy_property
+    def precision_matrix(self):
+        identity = torch.eye(
+            self._event_shape[-1],
+            device=self._unbroadcasted_scale_tril.device,
+            dtype=self._unbroadcasted_scale_tril.dtype,
+        )
+        return torch.cholesky_solve(identity, self._unbroadcasted_scale_tril).expand(
+            self._batch_shape + self._event_shape
+        )
+
+    @property
+    def mean(self):
+        return self.df.view(self._batch_shape + (1, 1)) * self.covariance_matrix
+
+    @property
+    def mode(self):
+        factor = self.df - self.covariance_matrix.shape[-1] - 1
+        factor[factor <= 0] = nan
+        return factor.view(self._batch_shape + (1, 1)) * self.covariance_matrix
+
+    @property
+    def variance(self):
+        V = self.covariance_matrix  # has shape (batch_shape x event_shape)
+        diag_V = V.diagonal(dim1=-2, dim2=-1)
+        return self.df.view(self._batch_shape + (1, 1)) * (
+            V.pow(2) + torch.einsum("...i,...j->...ij", diag_V, diag_V)
+        )
+
+    def _bartlett_sampling(self, sample_shape=torch.Size()):
+        p = self._event_shape[-1]  # has singleton shape
+
+        # Implemented Sampling using Bartlett decomposition
+        noise = _clamp_above_eps(
+            self._dist_chi2.rsample(sample_shape).sqrt()
+        ).diag_embed(dim1=-2, dim2=-1)
+
+        i, j = torch.tril_indices(p, p, offset=-1)
+        noise[..., i, j] = torch.randn(
+            torch.Size(sample_shape) + self._batch_shape + (int(p * (p - 1) / 2),),
+            dtype=noise.dtype,
+            device=noise.device,
+        )
+        chol = self._unbroadcasted_scale_tril @ noise
+        return chol @ chol.transpose(-2, -1)
+
+    def rsample(
+        self, sample_shape: _size = torch.Size(), max_try_correction=None
+    ) -> torch.Tensor:
+        r"""
+        .. warning::
+            In some cases, sampling algorithm based on Bartlett decomposition may return singular matrix samples.
+            Several tries to correct singular samples are performed by default, but it may end up returning
+            singular matrix samples. Singular samples may return `-inf` values in `.log_prob()`.
+            In those cases, the user should validate the samples and either fix the value of `df`
+            or adjust `max_try_correction` value for argument in `.rsample` accordingly.
+        """
+
+        if max_try_correction is None:
+            max_try_correction = 3 if torch._C._get_tracing_state() else 10
+
+        sample_shape = torch.Size(sample_shape)
+        sample = self._bartlett_sampling(sample_shape)
+
+        # Below part is to improve numerical stability temporally and should be removed in the future
+        is_singular = self.support.check(sample)
+        if self._batch_shape:
+            is_singular = is_singular.amax(self._batch_dims)
+
+        if torch._C._get_tracing_state():
+            # Less optimized version for JIT
+            for _ in range(max_try_correction):
+                sample_new = self._bartlett_sampling(sample_shape)
+                sample = torch.where(is_singular, sample_new, sample)
+
+                is_singular = ~self.support.check(sample)
+                if self._batch_shape:
+                    is_singular = is_singular.amax(self._batch_dims)
+
+        else:
+            # More optimized version with data-dependent control flow.
+            if is_singular.any():
+                warnings.warn("Singular sample detected.")
+
+                for _ in range(max_try_correction):
+                    sample_new = self._bartlett_sampling(is_singular[is_singular].shape)
+                    sample[is_singular] = sample_new
+
+                    is_singular_new = ~self.support.check(sample_new)
+                    if self._batch_shape:
+                        is_singular_new = is_singular_new.amax(self._batch_dims)
+                    is_singular[is_singular.clone()] = is_singular_new
+
+                    if not is_singular.any():
+                        break
+
+        return sample
+
+    def log_prob(self, value):
+        if self._validate_args:
+            self._validate_sample(value)
+        nu = self.df  # has shape (batch_shape)
+        p = self._event_shape[-1]  # has singleton shape
+        return (
+            -nu
+            * (
+                p * _log_2 / 2
+                + self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1)
+                .log()
+                .sum(-1)
+            )
+            - torch.mvlgamma(nu / 2, p=p)
+            + (nu - p - 1) / 2 * torch.linalg.slogdet(value).logabsdet
+            - torch.cholesky_solve(value, self._unbroadcasted_scale_tril)
+            .diagonal(dim1=-2, dim2=-1)
+            .sum(dim=-1)
+            / 2
+        )
+
+    def entropy(self):
+        nu = self.df  # has shape (batch_shape)
+        p = self._event_shape[-1]  # has singleton shape
+        V = self.covariance_matrix  # has shape (batch_shape x event_shape)
+        return (
+            (p + 1)
+            * (
+                p * _log_2 / 2
+                + self._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1)
+                .log()
+                .sum(-1)
+            )
+            + torch.mvlgamma(nu / 2, p=p)
+            - (nu - p - 1) / 2 * _mvdigamma(nu / 2, p=p)
+            + nu * p / 2
+        )
+
+    @property
+    def _natural_params(self):
+        nu = self.df  # has shape (batch_shape)
+        p = self._event_shape[-1]  # has singleton shape
+        return -self.precision_matrix / 2, (nu - p - 1) / 2
+
+    def _log_normalizer(self, x, y):
+        p = self._event_shape[-1]
+        return (y + (p + 1) / 2) * (
+            -torch.linalg.slogdet(-2 * x).logabsdet + _log_2 * p
+        ) + torch.mvlgamma(y + (p + 1) / 2, p=p)