text: # Copyright 2018 The TensorFlow Probability Authors.
  #
  # Licensed under the Apache License, Version 2.0 (the "License");
  # you may not use this file except in compliance with the License.
  # You may obtain a copy of the License at
  #
  #     http://www.apache.org/licenses/LICENSE-2.0
  #
  # Unless required by applicable law or agreed to in writing, software
  # distributed under the License is distributed on an "AS IS" BASIS,
  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  # See the License for the specific language governing permissions and
  # limitations under the License.
  # ============================================================================
  """Base classes for probability distributions."""
  
  from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  
  import abc
  import collections
  import contextlib
  import functools
  import inspect
  import types
  import decorator
  import numpy as np
  import six
  import tensorflow.compat.v2 as tf
  
  from tensorflow_probability.python.distributions import kullback_leibler
  from tensorflow_probability.python.distributions.internal import slicing
  from tensorflow_probability.python.internal import assert_util
  from tensorflow_probability.python.internal import auto_composite_tensor
  from tensorflow_probability.python.internal import distribution_util
  from tensorflow_probability.python.internal import dtype_util
  from tensorflow_probability.python.internal import name_util
  from tensorflow_probability.python.internal import nest_util
  from tensorflow_probability.python.internal import prefer_static as ps
  from tensorflow_probability.python.internal import tensorshape_util
  # Symbol import needed to avoid BUILD-dependency cycle
  from tensorflow_probability.python.math.generic import log1mexp
  from tensorflow.python.util import deprecation  # pylint: disable=g-direct-tensorflow-import
  from tensorflow.python.util import nest  # pylint: disable=g-direct-tensorflow-import
  from tensorflow.python.util import tf_inspect  # pylint: disable=g-direct-tensorflow-import
  
  
  __all__ = [
      'Distribution',
  ]
  
  _DISTRIBUTION_PUBLIC_METHOD_WRAPPERS = [
      'batch_shape',
      'batch_shape_tensor',
      'cdf',
      'covariance',
      'cross_entropy',
      'entropy',
      'event_shape',
      'event_shape_tensor',
      'kl_divergence',
      'log_cdf',
      'log_prob',
      'log_survival_function',
      'mean',
      'mode',
      'prob',
      'sample',
      'stddev',
      'survival_function',
      'variance',
  ]
  
  
  _ALWAYS_COPY_PUBLIC_METHOD_WRAPPERS = ['kl_divergence', 'cross_entropy']
  
  
  UNSET_VALUE = object()
  
  
  JAX_MODE = False  # Overwritten by rewrite script.
  
  
  @six.add_metaclass(abc.ABCMeta)
  class _BaseDistribution(tf.Module):
    """Abstract base class needed for resolving subclass hierarchy."""
    pass
  
  
  def _copy_fn(fn):
    """Create a deep copy of fn.
  
    Args:
      fn: a callable
  
    Returns:
      A `FunctionType`: a deep copy of fn.
  
    Raises:
      TypeError: if `fn` is not a callable.
    """
    if not callable(fn):
      raise TypeError('fn is not callable: {}'.format(fn))
    # The blessed way to copy a function. copy.deepcopy fails to create a
    # non-reference copy. Since:
    #   types.FunctionType == type(lambda: None),
    # and the docstring for the function type states:
    #
    #   function(code, globals[, name[, argdefs[, closure]]])
    #
    #   Create a function object from a code object and a dictionary.
    #   ...
    #
    # Here we can use this to create a new function with the old function's
    # code, globals, closure, etc.
    return types.FunctionType(
        code=fn.__code__, globals=fn.__globals__,
        name=fn.__name__, argdefs=fn.__defaults__,
        closure=fn.__closure__)
  
  
  def _update_docstring(old_str, append_str):
    """Update old_str by inserting append_str just before the 'Args:' section."""
    old_str = old_str or ''
    old_str_lines = old_str.split('\n')
  
    # Step 0: Prepend spaces to all lines of append_str. This is
    # necessary for correct markdown generation.
    append_str = '\n'.join('    %s' % line for line in append_str.split('\n'))
  
    # Step 1: Find mention of 'Args':
    has_args_ix = [
        ix for ix, line in enumerate(old_str_lines)
        if line.strip().lower() == 'args:']
    if has_args_ix:
      final_args_ix = has_args_ix[-1]
      return ('\n'.join(old_str_lines[:final_args_ix])
              + '\n\n' + append_str + '\n\n'
              + '\n'.join(old_str_lines[final_args_ix:]))
    else:
      return old_str + '\n\n' + append_str
  
  
  def _remove_dict_keys_with_value(dict_, val):
    """Removes `dict` keys which have have `self` as value."""
    return {k: v for k, v in dict_.items() if v is not val}
  
  
  def _cast_structure(value, structure):
    """Cast a structure."""
    if tf.nest.is_nested(structure):
      if nest._is_namedtuple(structure):  # pylint: disable=protected-access
        return type(structure)(*value)
      else:
        return type(structure)(value)
    return value
  
  
  def _set_sample_static_shape_for_tensor(x,
                                          event_shape,
                                          batch_shape,
                                          sample_shape):
    """Helper to `_set_sample_static_shape`; sets shape info for a `Tensor`."""
    sample_shape = tf.TensorShape(tf.get_static_value(sample_shape))
  
    ndims = tensorshape_util.rank(x.shape)
    sample_ndims = tensorshape_util.rank(sample_shape)
    batch_ndims = tensorshape_util.rank(batch_shape)
    event_ndims = tensorshape_util.rank(event_shape)
  
    # Infer rank(x).
    if (ndims is None and
        sample_ndims is not None and
        batch_ndims is not None and
        event_ndims is not None):
      ndims = sample_ndims + batch_ndims + event_ndims
      tensorshape_util.set_shape(x, [None] * ndims)
  
    # Infer sample shape.
    if ndims is not None and sample_ndims is not None:
      shape = tensorshape_util.concatenate(sample_shape,
                                           [None] * (ndims - sample_ndims))
      tensorshape_util.set_shape(x, shape)
  
    # Infer event shape.
    if ndims is not None and event_ndims is not None:
      shape = tf.TensorShape(
          [None]*(ndims - event_ndims)).concatenate(event_shape)
      tensorshape_util.set_shape(x, shape)
  
    # Infer batch shape.
    if batch_ndims is not None:
      if ndims is not None:
        if sample_ndims is None and event_ndims is not None:
          sample_ndims = ndims - batch_ndims - event_ndims
        elif event_ndims is None and sample_ndims is not None:
          event_ndims = ndims - batch_ndims - sample_ndims
      if sample_ndims is not None and event_ndims is not None:
        shape = tf.TensorShape([None]*sample_ndims).concatenate(
            batch_shape).concatenate([None]*event_ndims)
        tensorshape_util.set_shape(x, shape)
    return x
  
  
  class _DistributionMeta(abc.ABCMeta):
    """Helper metaclass for tfp.Distribution."""
  
    def __new__(mcs, classname, baseclasses, attrs):
      """Control the creation of subclasses of the Distribution class.
  
      The main purpose of this method is to properly propagate docstrings
      from private Distribution methods, like `_log_prob`, into their
      public wrappers as inherited by the Distribution base class
      (e.g. `log_prob`).
  
      Args:
        classname: The name of the subclass being created.
        baseclasses: A tuple of parent classes.
        attrs: A dict mapping new attributes to their values.
  
      Returns:
        The class object.
  
      Raises:
        TypeError: If `Distribution` is not a subclass of `BaseDistribution`, or
          the new class is derived via multiple inheritance and the first
          parent class is not a subclass of `BaseDistribution`.
        AttributeError:  If `Distribution` does not implement e.g. `log_prob`.
        ValueError:  If a `Distribution` public method lacks a docstring.
      """
      if not baseclasses:  # Nothing to be done for Distribution
        raise TypeError('Expected non-empty baseclass. Does Distribution '
                        'not subclass _BaseDistribution?')
      which_base = [
          base for base in baseclasses
          if base == _BaseDistribution or issubclass(base, Distribution)]
      base = which_base[0] if which_base else None
      if base is None or base == _BaseDistribution:
        # Nothing to be done for Distribution or unrelated subclass.
        return super(_DistributionMeta, mcs).__new__(
            mcs, classname, baseclasses, attrs)
      if not issubclass(base, Distribution):
        raise TypeError('First parent class declared for {} must be '
                        'Distribution, but saw "{}"'.format(
                            classname, base.__name__))
      for attr in _DISTRIBUTION_PUBLIC_METHOD_WRAPPERS:
        if attr in attrs:
          # The method is being overridden, do not update its docstring.
          continue
        special_attr = '_{}'.format(attr)
        class_attr_value = attrs.get(attr, None)
        base_attr_value = getattr(base, attr, None)
        if not base_attr_value:
          raise AttributeError(
              'Internal error: expected base class "{}" to '
              'implement method "{}"'.format(base.__name__, attr))
        class_special_attr_value = attrs.get(special_attr, None)
        class_special_attr_docstring = (
            None if class_special_attr_value is None else
            tf_inspect.getdoc(class_special_attr_value))
        if (class_special_attr_docstring or
            attr in _ALWAYS_COPY_PUBLIC_METHOD_WRAPPERS):
          class_attr_value = _copy_fn(base_attr_value)
          attrs[attr] = class_attr_value
  
        if not class_special_attr_docstring:
          # No docstring to append.
          continue
        class_attr_docstring = tf_inspect.getdoc(base_attr_value)
        if class_attr_docstring is None:
          raise ValueError(
              'Expected base class fn to contain a docstring: {}.{}'.format(
                  base.__name__, attr))
        class_attr_value.__doc__ = _update_docstring(
            class_attr_value.__doc__,
            'Additional documentation from `{}`:\n\n{}'.format(
                classname, class_special_attr_docstring))
  
      # Now we'll intercept the default __init__ if it exists.
      default_init = attrs.get('__init__', None)
      if default_init is None:
        # The class has no __init__ because its abstract. (And we won't add one.)
        return super(_DistributionMeta, mcs).__new__(
            mcs, classname, baseclasses, attrs)
  
      # Subclasses shouldn't inherit their parents' `_parameter_properties`,
      # since (in general) they'll have different parameters. Exceptions (for
      # convenience) are:
      #  - Subclasses that don't define their own `__init__` (handled above by
      #    the short-circuit when `default_init is None`).
      #  - Subclasses that define a passthrough `__init__(self, *args, **kwargs)`.
      #  - Direct children of `Distribution`, since the inherited method just
      #    raises a NotImplementedError.
      init_argspec = tf_inspect.getfullargspec(default_init)
      if ('_parameter_properties' not in attrs
          and base != Distribution
          # Passthrough exception: may only take `self` and at least one of
          # `*args` and `**kwargs`.
          and (len(init_argspec.args) > 1
               or not (init_argspec.varargs or init_argspec.varkw))):
        # TODO(b/183457779) remove warning and raise `NotImplementedError`.
        attrs['_parameter_properties'] = deprecation.deprecated(
            date='2021-07-01',
            instructions="""
  Calling `_parameter_properties` on subclass {classname} that redefines the
  parent ({basename}) `__init__` is unsafe and will raise an error in the future.
  Please implement an explicit `_parameter_properties` for the subclass. If the
  subclass `__init__` takes the same parameters as the parent, you may use the
  placeholder implementation:
  
    @classmethod
    def _parameter_properties(cls, dtype, num_classes=None):
      return {basename}._parameter_properties(
          dtype=dtype, num_classes=num_classes)
  
  """.format(classname=classname,
             basename=base.__name__))(base._parameter_properties)
  
      # pylint: disable=protected-access
      # For a comparison of different methods for wrapping functions, see:
      # https://hynek.me/articles/decorators/
      @decorator.decorator
      def wrapped_init(wrapped, self_, *args, **kwargs):
        """A 'master `__init__`' which is always called."""
        # We can't use `wrapped` because it results in a self reference which
        # confounds `tf.function`.
        del wrapped
        # Note: if we ever want to have things set in `self` before `__init__` is
        # called, here is the place to do it.
        self_._parameters = None
        default_init(self_, *args, **kwargs)
        # Note: if we ever want to override things set in `self` by subclass
        # `__init__`, here is the place to do it.
        if self_._parameters is None:
          # We prefer subclasses will set `parameters = dict(locals())` because
          # this has nearly zero overhead. However, failing to do this, we will
          # resolve the input arguments dynamically and only when needed.
          dummy_self = tuple()
          self_._parameters = self_._no_dependency(lambda: (  # pylint: disable=g-long-lambda
              _remove_dict_keys_with_value(
                  inspect.getcallargs(default_init, dummy_self, *args, **kwargs),
                  dummy_self)))
        elif hasattr(self_._parameters, 'pop'):
          self_._parameters = self_._no_dependency(
              _remove_dict_keys_with_value(self_._parameters, self_))
      # pylint: enable=protected-access
  
      attrs['__init__'] = wrapped_init(default_init)  # pylint: disable=no-value-for-parameter,assignment-from-no-return
      return super(_DistributionMeta, mcs).__new__(
          mcs, classname, baseclasses, attrs)
  
    def __init__(cls, name, bases, dct):
      super(_DistributionMeta, cls).__init__(name, bases, dct)
      if not JAX_MODE:
        return
      def flatten(dist):
        param_names = set(dist._composite_tensor_nonshape_params)  # pylint: disable=protected-access
        components = {param_name: getattr(
            dist, param_name, value) for param_name, value
                      in dist.parameters.items() if param_name in param_names}
        metadata = {param_name: value for param_name, value
                    in dist.parameters.items() if param_name not in param_names}
        if components:
          keys, values = zip(*sorted(components.items()))
        else:
          keys, values = (), ()
        # Mimics the logic in `tfp.experimental.composite_tensor` where we
        # aggressively try to convert arguments into Tensors.
        def _maybe_convert_to_tensor(value, name):
          try:
            value = tf.convert_to_tensor(value, name=name)
          except (ValueError, TypeError, AssertionError):
            pass
          return value
        values = tuple([_maybe_convert_to_tensor(value, name) for value, name,
                        in zip(values, keys)])
        return values, (keys, metadata)
      def unflatten(info, xs):
        keys, metadata = info
        parameters = dict(list(zip(keys, xs)), **metadata)
        return cls(**parameters)
      from jax import tree_util  # pylint: disable=g-import-not-at-top
      tree_util.register_pytree_node(cls, flatten, unflatten)
  
  
  @six.add_metaclass(_DistributionMeta)
  class Distribution(_BaseDistribution):
    """A generic probability distribution base class.
  
    `Distribution` is a base class for constructing and organizing properties
    (e.g., mean, variance) of random variables (e.g, Bernoulli, Gaussian).
  
    #### Subclassing
  
    Subclasses are expected to implement a leading-underscore version of the
    same-named function. The argument signature should be identical except for
    the omission of `name='...'`. For example, to enable `log_prob(value,
    name='log_prob')` a subclass should implement `_log_prob(value)`.
  
    Subclasses can append to public-level docstrings by providing
    docstrings for their method specializations. For example:
  
    ```python
    @distribution_util.AppendDocstring('Some other details.')
    def _log_prob(self, value):
      ...
    ```
  
    would add the string "Some other details." to the `log_prob` function
    docstring. This is implemented as a simple decorator to avoid python
    linter complaining about missing Args/Returns/Raises sections in the
    partial docstrings.
  
    TFP methods generally assume that Distribution subclasses implement at least
    the following methods:
    - `_sample_n`.
    - `_log_prob` or `_prob`.
    - `_event_shape` and `_event_shape_tensor`.
    - `_batch_shape` and `_batch_shape_tensor`.
  
    Some functionality may depend on implementing additional methods. It is common
    for Distribution subclasses to implement:
  
    - Relevant statistics, such as `_mean`, `_mode`, `_variance` and/or `_stddev`.
    - At least one of `_log_cdf`, `_cdf`, `_survival_function`, or
      `_log_survival_function`.
    - `_quantile`.
    - `_entropy`.
    - `_default_event_space_bijector`.
    - `_parameter_properties` (to support batch slicing and other features).
  
    Note that subclasses of existing Distributions that redefine `__init__` do
    *not* automatically inherit
    `_parameter_properties` annotations from their parent: the subclass must
    explicitly implement its own `_parameter_properties` method to support the
    features, such as batch slicing, that this enables.
  
    #### Broadcasting, batching, and shapes
  
    All distributions support batches of independent distributions of that type.
    The batch shape is determined by broadcasting together the parameters.
  
    The shape of arguments to `__init__`, `cdf`, `log_cdf`, `prob`, and
    `log_prob` reflect this broadcasting, as does the return value of `sample`.
  
    `sample_n_shape = [n] + batch_shape + event_shape`, where `sample_n_shape` is
    the shape of the `Tensor` returned from `sample(n)`, `n` is the number of
    samples, `batch_shape` defines how many independent distributions there are,
    and `event_shape` defines the shape of samples from each of those independent
    distributions. Samples are independent along the `batch_shape` dimensions, but
    not necessarily so along the `event_shape` dimensions (depending on the
    particulars of the underlying distribution).
  
    Using the `Uniform` distribution as an example:
  
    ```python
    minval = 3.0
    maxval = [[4.0, 6.0],
              [10.0, 12.0]]
  
    # Broadcasting:
    # This instance represents 4 Uniform distributions. Each has a lower bound at
    # 3.0 as the `minval` parameter was broadcasted to match `maxval`'s shape.
    u = Uniform(minval, maxval)
  
    # `event_shape` is `TensorShape([])`.
    event_shape = u.event_shape
    # `event_shape_t` is a `Tensor` which will evaluate to [].
    event_shape_t = u.event_shape_tensor()
  
    # Sampling returns a sample per distribution. `samples` has shape
    # [5, 2, 2], which is [n] + batch_shape + event_shape, where n=5,
    # batch_shape=[2, 2], and event_shape=[].
    samples = u.sample(5)
  
    # The broadcasting holds across methods. Here we use `cdf` as an example. The
    # same holds for `log_cdf` and the likelihood functions.
  
    # `cum_prob` has shape [2, 2] as the `value` argument was broadcasted to the
    # shape of the `Uniform` instance.
    cum_prob_broadcast = u.cdf(4.0)
  
    # `cum_prob`'s shape is [2, 2], one per distribution. No broadcasting
    # occurred.
    cum_prob_per_dist = u.cdf([[4.0, 5.0],
                               [6.0, 7.0]])
  
    # INVALID as the `value` argument is not broadcastable to the distribution's
    # shape.
    cum_prob_invalid = u.cdf([4.0, 5.0, 6.0])
    ```
  
    #### Shapes
  
    There are three important concepts associated with TensorFlow Distributions
    shapes:
  
    - Event shape describes the shape of a single draw from the distribution;
      it may be dependent across dimensions. For scalar distributions, the event
      shape is `[]`. For a 5-dimensional MultivariateNormal, the event shape is
      `[5]`.
    - Batch shape describes independent, not identically distributed draws, aka a
      "collection" or "bunch" of distributions.
    - Sample shape describes independent, identically distributed draws of batches
      from the distribution family.
  
    The event shape and the batch shape are properties of a Distribution object,
    whereas the sample shape is associated with a specific call to `sample` or
    `log_prob`.
  
    For detailed usage examples of TensorFlow Distributions shapes, see
    [this tutorial](
    https://github.com/tensorflow/probability/blob/master/tensorflow_probability/examples/jupyter_notebooks/Understanding_TensorFlow_Distributions_Shapes.ipynb)
  
    #### Parameter values leading to undefined statistics or distributions.
  
    Some distributions do not have well-defined statistics for all initialization
    parameter values. For example, the beta distribution is parameterized by
    positive real numbers `concentration1` and `concentration0`, and does not have
    well-defined mode if `concentration1 < 1` or `concentration0 < 1`.
  
    The user is given the option of raising an exception or returning `NaN`.
  
    ```python
    a = tf.exp(tf.matmul(logits, weights_a))
    b = tf.exp(tf.matmul(logits, weights_b))
  
    # Will raise exception if ANY batch member has a < 1 or b < 1.
    dist = distributions.beta(a, b, allow_nan_stats=False)
    mode = dist.mode()
  
    # Will return NaN for batch members with either a < 1 or b < 1.
    dist = distributions.beta(a, b, allow_nan_stats=True)  # Default behavior
    mode = dist.mode()
    ```
  
    In all cases, an exception is raised if *invalid* parameters are passed, e.g.
  
    ```python
    # Will raise an exception if any Op is run.
    negative_a = -1.0 * a  # beta distribution by definition has a > 0.
    dist = distributions.beta(negative_a, b, allow_nan_stats=True)
    dist.mean()
    ```
  
    """
  
    def __init__(self,
                 dtype,
                 reparameterization_type,
                 validate_args,
                 allow_nan_stats,
                 parameters=None,
                 graph_parents=None,
                 name=None):
      """Constructs the `Distribution`.
  
      **This is a private method for subclass use.**
  
      Args:
        dtype: The type of the event samples. `None` implies no type-enforcement.
        reparameterization_type: Instance of `ReparameterizationType`.
          If `tfd.FULLY_REPARAMETERIZED`, then samples from the distribution are
          fully reparameterized, and straight-through gradients are supported.
          If `tfd.NOT_REPARAMETERIZED`, then samples from the distribution are not
          fully reparameterized, and straight-through gradients are either
          partially unsupported or are not supported at all.
        validate_args: Python `bool`, default `False`. When `True` distribution
          parameters are checked for validity despite possibly degrading runtime
          performance. When `False` invalid inputs may silently render incorrect
          outputs.
        allow_nan_stats: Python `bool`, default `True`. When `True`, statistics
          (e.g., mean, mode, variance) use the value "`NaN`" to indicate the
          result is undefined. When `False`, an exception is raised if one or
          more of the statistic's batch members are undefined.
        parameters: Python `dict` of parameters used to instantiate this
          `Distribution`.
        graph_parents: Python `list` of graph prerequisites of this
          `Distribution`.
        name: Python `str` name prefixed to Ops created by this class. Default:
          subclass name.
  
      Raises:
        ValueError: if any member of graph_parents is `None` or not a `Tensor`.
      """
      if not name:
        name = type(self).__name__
        name = name_util.camel_to_lower_snake(name)
      name = name_util.get_name_scope_name(name)
      name = name_util.strip_invalid_chars(name)
      super(Distribution, self).__init__(name=name)
      self._name = name
  
      graph_parents = [] if graph_parents is None else graph_parents
      for i, t in enumerate(graph_parents):
        if t is None or not tf.is_tensor(t):
          raise ValueError('Graph parent item %d is not a Tensor; %s.' % (i, t))
      self._dtype = self._no_dependency(dtype)
      self._reparameterization_type = reparameterization_type
      self._allow_nan_stats = allow_nan_stats
      self._validate_args = validate_args
      self._parameters = self._no_dependency(parameters)
      self._parameters_sanitized = False
      self._graph_parents = graph_parents
      self._defer_all_assertions = (
          auto_composite_tensor.is_deferred_assertion_context())
  
      if not self._defer_all_assertions:
        self._initial_parameter_control_dependencies = tuple(
            d for d in self._parameter_control_dependencies(is_init=True)
            if d is not None)
      else:
        self._initial_parameter_control_dependencies = ()
  
      if self._initial_parameter_control_dependencies:
        self._initial_parameter_control_dependencies = (
            tf.group(*self._initial_parameter_control_dependencies),)
  
    @property
    def _composite_tensor_params(self):
      """A tuple describing which parameters are expected to be tensors.
  
      CompositeTensor requires us to partition dynamic (tensor) parts from static
      (metadata) parts like 'validate_args'.  This collects the keys of parameters
      which are expected to be tensors.
      """
      return (self._composite_tensor_nonshape_params +
              self._composite_tensor_shape_params)
  
    @property
    def _composite_tensor_nonshape_params(self):
      """A tuple describing which parameters are non-shape-related tensors.
  
      Flattening in JAX involves many of the same considerations with regards to
      identifying tensor arguments for the purposes of CompositeTensor, except
      that shape-related items will be considered metadata.  This property
      identifies the keys of parameters that are expected to be tensors, except
      those that are shape-related.
      """
      return tuple(self._params_event_ndims().keys())
  
    @property
    def _composite_tensor_shape_params(self):
      """A tuple describing which parameters are shape-related tensors.
  
      Flattening in JAX involves many of the same considerations with regards to
      identifying tensor arguments for the purposes of CompositeTensor, except
      that shape-related items will be considered metadata.  This property
      identifies the keys of parameters that are expected to be shape-related
      tensors, so that they can be collected appropriately in CompositeTensor but
      not in JAX applications.
      """
      return ()
  
    @classmethod
    def _parameter_properties(cls, dtype, num_classes=None):
      raise NotImplementedError(
          '_parameter_properties` is not implemented: {}. '
          'Note that subclasses that redefine `__init__` are not assumed to '
          'share parameters with their parent class and must provide a separate '
          'implementation.'.format(cls.__name__))
  
    @classmethod
    def parameter_properties(cls, dtype=tf.float32, num_classes=None):
      """Returns a dict mapping constructor arg names to property annotations.
  
      This dict should include an entry for each of the distribution's
      `Tensor`-valued constructor arguments.
  
      Distribution subclasses are not required to implement
      `_parameter_properties`, so this method may raise `NotImplementedError`.
      Providing a `_parameter_properties` implementation enables several advanced
      features, including:
        - Distribution batch slicing (`sliced_distribution = distribution[i:j]`).
        - Automatic inference of `_batch_shape` and
          `_batch_shape_tensor`, which must otherwise be computed explicitly.
        - Automatic instantiation of the distribution within TFP's internal
          property tests.
        - Automatic construction of 'trainable' instances of the distribution
          using appropriate bijectors to avoid violating parameter constraints.
          This enables the distribution family to be used easily as a
          surrogate posterior in variational inference.
  
      In the future, parameter property annotations may enable additional
      functionality; for example, returning Distribution instances from
      `tf.vectorized_map`.
  
      Args:
        dtype: Optional float `dtype` to assume for continuous-valued parameters.
          Some constraining bijectors require advance knowledge of the dtype
          because certain constants (e.g., `tfb.Softplus.low`) must be
          instantiated with the same dtype as the values to be transformed.
        num_classes: Optional `int` `Tensor` number of classes to assume when
          inferring the shape of parameters for categorical-like distributions.
          Otherwise ignored.
  
      Returns:
        parameter_properties: A
          `str -> `tfp.python.internal.parameter_properties.ParameterProperties`
          dict mapping constructor argument names to `ParameterProperties`
          instances.
      Raises:
        NotImplementedError: if the distribution class does not implement
          `_parameter_properties`.
      """
      with tf.name_scope('parameter_properties'):
        return cls._parameter_properties(dtype, num_classes=num_classes)
  
    @classmethod
    @deprecation.deprecated('2021-03-01',
                            'The `param_shapes` method of `tfd.Distribution` is '
                            'deprecated; use `parameter_properties` instead.')
    def param_shapes(cls, sample_shape, name='DistributionParamShapes'):
      """Shapes of parameters given the desired shape of a call to `sample()`.
  
      This is a class method that describes what key/value arguments are required
      to instantiate the given `Distribution` so that a particular shape is
      returned for that instance's call to `sample()`.
  
      Subclasses should override class method `_param_shapes`.
  
      Args:
        sample_shape: `Tensor` or python list/tuple. Desired shape of a call to
          `sample()`.
        name: name to prepend ops with.
  
      Returns:
        `dict` of parameter name to `Tensor` shapes.
      """
      with tf.name_scope(name):
        param_shapes = {}
        for (param_name, param) in cls.parameter_properties().items():
          param_shapes[param_name] = tf.convert_to_tensor(
              param.shape_fn(sample_shape), dtype=tf.int32)
        return param_shapes
  
    @classmethod
    @deprecation.deprecated(
        '2021-03-01', 'The `param_static_shapes` method of `tfd.Distribution` is '
        'deprecated; use `parameter_properties` instead.')
    def param_static_shapes(cls, sample_shape):
      """param_shapes with static (i.e. `TensorShape`) shapes.
  
      This is a class method that describes what key/value arguments are required
      to instantiate the given `Distribution` so that a particular shape is
      returned for that instance's call to `sample()`. Assumes that the sample's
      shape is known statically.
  
      Subclasses should override class method `_param_shapes` to return
      constant-valued tensors when constant values are fed.
  
      Args:
        sample_shape: `TensorShape` or python list/tuple. Desired shape of a call
          to `sample()`.
  
      Returns:
        `dict` of parameter name to `TensorShape`.
  
      Raises:
        ValueError: if `sample_shape` is a `TensorShape` and is not fully defined.
      """
      if isinstance(sample_shape, tf.TensorShape):
        if not tensorshape_util.is_fully_defined(sample_shape):
          raise ValueError('TensorShape sample_shape must be fully defined')
        sample_shape = tensorshape_util.as_list(sample_shape)
  
      params = cls.param_shapes(sample_shape)
  
      static_params = {}
      for name, shape in params.items():
        static_shape = tf.get_static_value(shape)
        if static_shape is None:
          raise ValueError(
              'sample_shape must be a fully-defined TensorShape or list/tuple')
        static_params[name] = tf.TensorShape(static_shape)
  
      return static_params
  
    @property
    def name(self):
      """Name prepended to all ops created by this `Distribution`."""
      return self._name if hasattr(self, '_name') else None
  
    @property
    def dtype(self):
      """The `DType` of `Tensor`s handled by this `Distribution`."""
      return self._dtype if hasattr(self, '_dtype') else None
  
    @property
    def parameters(self):
      """Dictionary of parameters used to instantiate this `Distribution`."""
      # Remove 'self', '__class__', or other special variables. These can appear
      # if the subclass used: `parameters = dict(locals())`.
      if (not hasattr(self, '_parameters_sanitized') or
          not self._parameters_sanitized):
        p = self._parameters() if callable(self._parameters) else self._parameters
        self._parameters = self._no_dependency({
            k: v for k, v in p.items()
            if not k.startswith('__') and v is not self})
        self._parameters_sanitized = True
      # In some situations, the Distribution metaclass logic defers the evaluation
      # of parameters, but at this point we actually want to evaluate the
      # parameters.
      return dict(
          self._parameters() if callable(self._parameters) else self._parameters)
  
    def _params_event_ndims(self):
      """Returns a dict mapping constructor argument names to per-event rank.
  
      The ranks are pulled from `cls.parameter_properties()`; this is a
      convenience wrapper.
  
      Returns:
        params_event_ndims: Per-event parameter ranks, a `str->int dict`.
      """
      try:
        return {
            k: param.instance_event_ndims(self)
            for (k, param) in type(self).parameter_properties().items()
            if param.is_tensor
        }
      except NotImplementedError:
        raise NotImplementedError(
            '{} does not support batch slicing; must implement '
            '_parameter_properties.'.format(type(self)))
  
    def __getitem__(self, slices):
      """Slices the batch axes of this distribution, returning a new instance.
  
      ```python
      b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
      b.batch_shape  # => [3, 5, 7, 9]
      b2 = b[:, tf.newaxis, ..., -2:, 1::2]
      b2.batch_shape  # => [3, 1, 5, 2, 4]
  
      x = tf.random.normal([5, 3, 2, 2])
      cov = tf.matmul(x, x, transpose_b=True)
      chol = tf.linalg.cholesky(cov)
      loc = tf.random.normal([4, 1, 3, 1])
      mvn = tfd.MultivariateNormalTriL(loc, chol)
      mvn.batch_shape  # => [4, 5, 3]
      mvn.event_shape  # => [2]
      mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
      mvn2.batch_shape  # => [4, 2, 3, 1]
      mvn2.event_shape  # => [2]
      ```
  
      Args:
        slices: slices from the [] operator
  
      Returns:
        dist: A new `tfd.Distribution` instance with sliced parameters.
      """
      return slicing.batch_slice(self, self._params_event_ndims(), {}, slices)
  
    def __iter__(self):
      raise TypeError('{!r} object is not iterable'.format(type(self).__name__))
  
    @property
    def reparameterization_type(self):
      """Describes how samples from the distribution are reparameterized.
  
      Currently this is one of the static instances
      `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.
  
      Returns:
        An instance of `ReparameterizationType`.
      """
      return self._reparameterization_type
  
    @property
    def allow_nan_stats(self):
      """Python `bool` describing behavior when a stat is undefined.
  
      Stats return +/- infinity when it makes sense. E.g., the variance of a
      Cauchy distribution is infinity. However, sometimes the statistic is
      undefined, e.g., if a distribution's pdf does not achieve a maximum within
      the support of the distribution, the mode is undefined. If the mean is
      undefined, then by definition the variance is undefined. E.g. the mean for
      Student's T for df = 1 is undefined (no clear way to say it is either + or -
      infinity), so the variance = E[(X - mean)**2] is also undefined.
  
      Returns:
        allow_nan_stats: Python `bool`.
      """
      return self._allow_nan_stats
  
    @property
    def validate_args(self):
      """Python `bool` indicating possibly expensive checks are enabled."""
      return self._validate_args
  
    @property
    def experimental_shard_axis_names(self):
      """The list or structure of lists of active shard axis names."""
      return []
  
    def copy(self, **override_parameters_kwargs):
      """Creates a deep copy of the distribution.
  
      Note: the copy distribution may continue to depend on the original
      initialization arguments.
  
      Args:
        **override_parameters_kwargs: String/value dictionary of initialization
          arguments to override with new values.
  
      Returns:
        distribution: A new instance of `type(self)` initialized from the union
          of self.parameters and override_parameters_kwargs, i.e.,
          `dict(self.parameters, **override_parameters_kwargs)`.
      """
      try:
        # We want track provenance from origin variables, so we use batch_slice
        # if this distribution supports slicing. See the comment on
        # PROVENANCE_ATTR in slicing.py
        return slicing.batch_slice(self, self._params_event_ndims(),
                                   override_parameters_kwargs, Ellipsis)
      except NotImplementedError:
        pass
      parameters = dict(self.parameters, **override_parameters_kwargs)
      d = type(self)(**parameters)
      # pylint: disable=protected-access
      d._parameters = self._no_dependency(parameters)
      d._parameters_sanitized = True
      # pylint: enable=protected-access
      return d
  
    def _inferred_batch_shape_tensor(self):
      """Infers batch shape from parameters.
  
      The overall batch shape is inferred by broadcasting the batch shapes of
      all parameters,
  
      ```python
      parameter_batch_shapes = []
      for name, properties in self.parameter_properties.items():
        parameter = self.parameters[name]
        parameter_batch_shapes.append(
          base_shape(parameter)[:-properties.instance_event_ndims(parameter)])
      ```
  
      where a parameter's `base_shape` is its batch shape if it
      defines one (e.g., if it is a Distribution, LinearOperator, etc.), and its
      Tensor shape otherwise. Parameters with structured batch shape
      (in particular, non-autobatched JointDistributions) are not currently
      supported.
  
      Returns:
        batch_shape_tensor: `Tensor` broadcast batch shape of all parameters.
      """
      # TODO(davmre): support parameters with structured batch shape, like
      # non-autobatched JDs, in cases where there's an 'obvious' semantics.
      # For example, if there's only one parameter, we can probably just pass
      # through its batch shape.
      batch_shapes = [[]]
      parameter_properties = type(self).parameter_properties()
      for param_name, param in self.parameters.items():
        if param is None:
          continue
        if param_name not in parameter_properties:
          continue
        properties = parameter_properties[param_name]
  
        ndims = properties.instance_event_ndims(self)
        batch_shapes += nest.flatten_up_to(
            ndims,
            nest.map_structure_up_to(
                ndims,
                lambda p, nd: _parameter_batch_shape_tensor(  # pylint: disable=g-long-lambda
                    base_shape=_get_base_shape_tensor(p),
                    event_ndims=nd),
                param,
                ndims))
      return functools.reduce(ps.broadcast_shape, batch_shapes)
  
    def _batch_shape_tensor(self):
      raise NotImplementedError(
          'batch_shape_tensor is not implemented: {}'.format(type(self).__name__))
  
    def batch_shape_tensor(self, name='batch_shape_tensor'):
      """Shape of a single sample from a single event index as a 1-D `Tensor`.
  
      The batch dimensions are indexes into independent, non-identical
      parameterizations of this distribution.
  
      Args:
        name: name to give to the op
  
      Returns:
        batch_shape: `Tensor`.
      """
      with self._name_and_control_scope(name):
        # Joint distributions may have a structured `batch shape_tensor` or a
        # single `batch_shape_tensor` that applies to all components. (Simple
        # distributions always have a single `batch_shape_tensor`.) If the
        # distribution's `batch_shape` is an instance of `tf.TensorShape`, we
        # infer that `batch_shape_tensor` is not structured.
        shallow_structure = (None if isinstance(self.batch_shape, tf.TensorShape)
                             else self.dtype)
        if all([tensorshape_util.is_fully_defined(s)
                for s in nest.flatten_up_to(
                    shallow_structure, self.batch_shape, check_types=False)]):
          batch_shape = nest.map_structure_up_to(
              shallow_structure,
              tensorshape_util.as_list,
              self.batch_shape, check_types=False)
        else:
          batch_shape = self._batch_shape_tensor()
  
        def conversion_fn(s):
          return tf.identity(
              tf.convert_to_tensor(s, dtype=tf.int32), name='batch_shape')
        if JAX_MODE:
          conversion_fn = ps.convert_to_shape_tensor
        return nest.map_structure_up_to(
            shallow_structure,
            conversion_fn,
            batch_shape, check_types=False)
  
    def _inferred_batch_shape(self):
      """Infers static batch shape from parameters.
  
      The overall batch shape is inferred by broadcasting the batch shapes of
      all parameters
  
      ```python
      parameter_batch_shapes = []
      for name, properties in self.parameter_properties.items():
        parameter = self.parameters[name]
        parameter_batch_shapes.append(
          base_shape(parameter)[:-properties.instance_event_ndims(parameter)])
      ```
  
      where a parameter's `base_shape` is its batch shape if it
      defines one (e.g., if it is a Distribution, LinearOperator, etc.), and its
      Tensor shape otherwise. Distributions with structured batch shape
      (in particular, non-autobatched JointDistributions) are not currently
      supported.
  
      Returns:
        batch_shape: `tf.TensorShape` broadcast batch shape of all parameters; may
          be partially defined or unknown.
      """
      batch_shapes = [tf.TensorShape([])]
      parameter_properties = type(self).parameter_properties()
      for param_name, param in self.parameters.items():
        if param is None:
          continue
        if param_name not in parameter_properties:
          continue
        properties = parameter_properties[param_name]
  
        # Note that `ndims` may be returned here as a Tensor, but
        # `_batch_shape_from_parameter` is smart about avoiding graph side effects
        # (returns `TensorShape(None)` if a static value is not available).
        ndims = properties.instance_event_ndims(self)
        batch_shapes += nest.flatten_up_to(
            ndims,
            nest.map_structure_up_to(
                ndims,
                lambda p, nd: _parameter_batch_shape(  # pylint: disable=g-long-lambda
                    base_shape=_get_base_shape(p),
                    event_ndims=nd),
                param,
                ndims))
      return functools.reduce(tf.broadcast_static_shape,
                              tf.nest.flatten(batch_shapes))
  
    def _batch_shape(self):
      return None
  
    @property
    def batch_shape(self):
      """Shape of a single sample from a single event index as a `TensorShape`.
  
      May be partially defined or unknown.
  
      The batch dimensions are indexes into independent, non-identical
      parameterizations of this distribution.
  
      Returns:
        batch_shape: `TensorShape`, possibly unknown.
      """
      batch_shape = self._batch_shape()
      # See comment in `batch_shape_tensor()` on structured batch shapes. If
      # `_batch_shape()` is a `tf.TensorShape` instance or a flat list/tuple that
      # does not contain `tf.TensorShape`s, we infer that it is not structured.
      if (isinstance(batch_shape, tf.TensorShape)
          or all(len(path) == 1 and not isinstance(s, tf.TensorShape)
                 for path, s in nest.flatten_with_tuple_paths(batch_shape))):
        return tf.TensorShape(batch_shape)
      return nest.map_structure_up_to(
          self.dtype, tf.TensorShape, batch_shape, check_types=False)
  
    def _event_shape_tensor(self):
      raise NotImplementedError(
          'event_shape_tensor is not implemented: {}'.format(type(self).__name__))
  
    def event_shape_tensor(self, name='event_shape_tensor'):
      """Shape of a single sample from a single batch as a 1-D int32 `Tensor`.
  
      Args:
        name: name to give to the op
  
      Returns:
        event_shape: `Tensor`.
      """
      with self._name_and_control_scope(name):
        if all([tensorshape_util.is_fully_defined(s)
                for s in nest.flatten(self.event_shape)]):
          event_shape = nest.map_structure_up_to(
              self.dtype,
              tensorshape_util.as_list,
              self.event_shape, check_types=False)
        else:
          event_shape = self._event_shape_tensor()
        def conversion_fn(s):
          return tf.identity(
              tf.convert_to_tensor(s, dtype=tf.int32), name='event_shape')
        if JAX_MODE:
          conversion_fn = ps.convert_to_shape_tensor
        return nest.map_structure_up_to(
            self.dtype,
            conversion_fn,
            event_shape, check_types=False)
  
    def _event_shape(self):
      return None
  
    @property
    def event_shape(self):
      """Shape of a single sample from a single batch as a `TensorShape`.
  
      May be partially defined or unknown.
  
      Returns:
        event_shape: `TensorShape`, possibly unknown.
      """
      return nest.map_structure_up_to(
          self.dtype, tf.TensorShape, self._event_shape(), check_types=False)
  
    def is_scalar_event(self, name='is_scalar_event'):
      """Indicates that `event_shape == []`.
  
      Args:
        name: Python `str` prepended to names of ops created by this function.
  
      Returns:
        is_scalar_event: `bool` scalar `Tensor`.
      """
      with self._name_and_control_scope(name):
        return tf.convert_to_tensor(
            self._is_scalar_helper(self.event_shape, self.event_shape_tensor),
            name='is_scalar_event')
  
    def is_scalar_batch(self, name='is_scalar_batch'):
      """Indicates that `batch_shape == []`.
  
      Args:
        name: Python `str` prepended to names of ops created by this function.
  
      Returns:
        is_scalar_batch: `bool` scalar `Tensor`.
      """
      with self._name_and_control_scope(name):
        return tf.convert_to_tensor(
            self._is_scalar_helper(self.batch_shape, self.batch_shape_tensor),
            name='is_scalar_batch')
  
    def _sample_n(self, n, seed=None, **kwargs):
      raise NotImplementedError('sample_n is not implemented: {}'.format(
          type(self).__name__))
  
    def _call_sample_n(self, sample_shape, seed, name, **kwargs):
      """Wrapper around _sample_n."""
      with self._name_and_control_scope(name):
        if JAX_MODE and seed is None:
          raise ValueError('Must provide JAX PRNGKey as `dist.sample(seed=.)`')
        sample_shape = ps.convert_to_shape_tensor(
            ps.cast(sample_shape, tf.int32), name='sample_shape')
        sample_shape, n = self._expand_sample_shape_to_vector(
            sample_shape, 'sample_shape')
        samples = self._sample_n(
            n, seed=seed() if callable(seed) else seed, **kwargs)
        batch_event_shape = ps.shape(samples)[1:]
        final_shape = ps.concat([sample_shape, batch_event_shape], 0)
        samples = tf.reshape(samples, final_shape)
        samples = self._set_sample_static_shape(samples, sample_shape)
        return samples
  
    def sample(self, sample_shape=(), seed=None, name='sample', **kwargs):
      """Generate samples of the specified shape.
  
      Note that a call to `sample()` without arguments will generate a single
      sample.
  
      Args:
        sample_shape: 0D or 1D `int32` `Tensor`. Shape of the generated samples.
        seed: Python integer or `tfp.util.SeedStream` instance, for seeding PRNG.
        name: name to give to the op.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        samples: a `Tensor` with prepended dimensions `sample_shape`.
      """
      return self._call_sample_n(sample_shape, seed, name, **kwargs)
  
    def _call_log_prob(self, value, name, **kwargs):
      """Wrapper around _log_prob."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype,
          allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        if hasattr(self, '_log_prob'):
          return self._log_prob(value, **kwargs)
        if hasattr(self, '_prob'):
          return tf.math.log(self._prob(value, **kwargs))
        raise NotImplementedError('log_prob is not implemented: {}'.format(
            type(self).__name__))
  
    def log_prob(self, value, name='log_prob', **kwargs):
      """Log probability density/mass function.
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        log_prob: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
          values of type `self.dtype`.
      """
      return self._call_log_prob(value, name, **kwargs)
  
    def _call_prob(self, value, name, **kwargs):
      """Wrapper around _prob."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype,
          allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        if hasattr(self, '_prob'):
          return self._prob(value, **kwargs)
        if hasattr(self, '_log_prob'):
          return tf.exp(self._log_prob(value, **kwargs))
        raise NotImplementedError('prob is not implemented: {}'.format(
            type(self).__name__))
  
    def prob(self, value, name='prob', **kwargs):
      """Probability density/mass function.
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        prob: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
          values of type `self.dtype`.
      """
      return self._call_prob(value, name, **kwargs)
  
    def _call_unnormalized_log_prob(self, value, name, **kwargs):
      """Wrapper around _unnormalized_log_prob."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype, allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        if hasattr(self, '_unnormalized_log_prob'):
          return self._unnormalized_log_prob(value, **kwargs)
        if hasattr(self, '_unnormalized_prob'):
          return tf.math.log(self._unnormalized_prob(value, **kwargs))
        if hasattr(self, '_log_prob'):
          return self._log_prob(value, **kwargs)
        if hasattr(self, '_prob'):
          return tf.math.log(self._prob(value, **kwargs))
        raise NotImplementedError(
            'unnormalized_log_prob is not implemented: {}'.format(
                type(self).__name__))
  
    def unnormalized_log_prob(self,
                              value,
                              name='unnormalized_log_prob',
                              **kwargs):
      """Potentially unnormalized log probability density/mass function.
  
      This function is similar to `log_prob`, but does not require that the
      return value be normalized.  (Normalization here refers to the total
      integral of probability being one, as it should be by definition for any
      probability distribution.)  This is useful, for example, for distributions
      where the normalization constant is difficult or expensive to compute.  By
      default, this simply calls `log_prob`.
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        unnormalized_log_prob: a `Tensor` of shape
          `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.
      """
      return self._call_unnormalized_log_prob(value, name, **kwargs)
  
    def _call_log_cdf(self, value, name, **kwargs):
      """Wrapper around _log_cdf."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype,
          allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        if hasattr(self, '_log_cdf'):
          return self._log_cdf(value, **kwargs)
        if hasattr(self, '_cdf'):
          return tf.math.log(self._cdf(value, **kwargs))
        raise NotImplementedError('log_cdf is not implemented: {}'.format(
            type(self).__name__))
  
    def log_cdf(self, value, name='log_cdf', **kwargs):
      """Log cumulative distribution function.
  
      Given random variable `X`, the cumulative distribution function `cdf` is:
  
      ```none
      log_cdf(x) := Log[ P[X <= x] ]
      ```
  
      Often, a numerical approximation can be used for `log_cdf(x)` that yields
      a more accurate answer than simply taking the logarithm of the `cdf` when
      `x << -1`.
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        logcdf: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
          values of type `self.dtype`.
      """
      return self._call_log_cdf(value, name, **kwargs)
  
    def _call_cdf(self, value, name, **kwargs):
      """Wrapper around _cdf."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype,
          allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        if hasattr(self, '_cdf'):
          return self._cdf(value, **kwargs)
        if hasattr(self, '_log_cdf'):
          return tf.exp(self._log_cdf(value, **kwargs))
        raise NotImplementedError('cdf is not implemented: {}'.format(
            type(self).__name__))
  
    def cdf(self, value, name='cdf', **kwargs):
      """Cumulative distribution function.
  
      Given random variable `X`, the cumulative distribution function `cdf` is:
  
      ```none
      cdf(x) := P[X <= x]
      ```
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        cdf: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
          values of type `self.dtype`.
      """
      return self._call_cdf(value, name, **kwargs)
  
    def _log_survival_function(self, value, **kwargs):
      raise NotImplementedError(
          'log_survival_function is not implemented: {}'.format(
              type(self).__name__))
  
    def _call_log_survival_function(self, value, name, **kwargs):
      """Wrapper around _log_survival_function."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype,
          allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        try:
          return self._log_survival_function(value, **kwargs)
        except NotImplementedError:
          if hasattr(self, '_log_cdf'):
            return log1mexp(self._log_cdf(value, **kwargs))
          if hasattr(self, '_cdf'):
            return tf.math.log1p(-self._cdf(value, **kwargs))
          raise
  
    def log_survival_function(self, value, name='log_survival_function',
                              **kwargs):
      """Log survival function.
  
      Given random variable `X`, the survival function is defined:
  
      ```none
      log_survival_function(x) = Log[ P[X > x] ]
                               = Log[ 1 - P[X <= x] ]
                               = Log[ 1 - cdf(x) ]
      ```
  
      Typically, different numerical approximations can be used for the log
      survival function, which are more accurate than `1 - cdf(x)` when `x >> 1`.
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type
          `self.dtype`.
      """
      return self._call_log_survival_function(value, name, **kwargs)
  
    def _survival_function(self, value, **kwargs):
      raise NotImplementedError('survival_function is not implemented: {}'.format(
          type(self).__name__))
  
    def _call_survival_function(self, value, name, **kwargs):
      """Wrapper around _survival_function."""
      value = _cast_structure(value, self.dtype)
      value = nest_util.convert_to_nested_tensor(
          value, name='value', dtype_hint=self.dtype,
          allow_packing=True)
      with self._name_and_control_scope(name, value, kwargs):
        try:
          return self._survival_function(value, **kwargs)
        except NotImplementedError:
          if hasattr(self, '_log_cdf'):
            return -tf.math.expm1(self._log_cdf(value, **kwargs))
          if hasattr(self, '_cdf'):
            return 1. - self.cdf(value, **kwargs)
          raise
  
    def survival_function(self, value, name='survival_function', **kwargs):
      """Survival function.
  
      Given random variable `X`, the survival function is defined:
  
      ```none
      survival_function(x) = P[X > x]
                           = 1 - P[X <= x]
                           = 1 - cdf(x).
      ```
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type
          `self.dtype`.
      """
      return self._call_survival_function(value, name, **kwargs)
  
    def _entropy(self, **kwargs):
      raise NotImplementedError('entropy is not implemented: {}'.format(
          type(self).__name__))
  
    def entropy(self, name='entropy', **kwargs):
      """Shannon entropy in nats."""
      with self._name_and_control_scope(name):
        return self._entropy(**kwargs)
  
    def _mean(self, **kwargs):
      raise NotImplementedError('mean is not implemented: {}'.format(
          type(self).__name__))
  
    def mean(self, name='mean', **kwargs):
      """Mean."""
      with self._name_and_control_scope(name):
        return self._mean(**kwargs)
  
    def _quantile(self, value, **kwargs):
      raise NotImplementedError('quantile is not implemented: {}'.format(
          type(self).__name__))
  
    def _call_quantile(self, value, name, **kwargs):
      with self._name_and_control_scope(name):
        dtype = tf.float32 if tf.nest.is_nested(self.dtype) else self.dtype
        value = tf.convert_to_tensor(value, name='value', dtype_hint=dtype)
        if self.validate_args:
          value = distribution_util.with_dependencies([
              assert_util.assert_less_equal(value, tf.cast(1, value.dtype),
                                            message='`value` must be <= 1'),
              assert_util.assert_greater_equal(value, tf.cast(0, value.dtype),
                                               message='`value` must be >= 0')
          ], value)
        return self._quantile(value, **kwargs)
  
    def quantile(self, value, name='quantile', **kwargs):
      """Quantile function. Aka 'inverse cdf' or 'percent point function'.
  
      Given random variable `X` and `p in [0, 1]`, the `quantile` is:
  
      ```none
      quantile(p) := x such that P[X <= x] == p
      ```
  
      Args:
        value: `float` or `double` `Tensor`.
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        quantile: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
          values of type `self.dtype`.
      """
      return self._call_quantile(value, name, **kwargs)
  
    def _variance(self, **kwargs):
      raise NotImplementedError('variance is not implemented: {}'.format(
          type(self).__name__))
  
    def variance(self, name='variance', **kwargs):
      """Variance.
  
      Variance is defined as,
  
      ```none
      Var = E[(X - E[X])**2]
      ```
  
      where `X` is the random variable associated with this distribution, `E`
      denotes expectation, and `Var.shape = batch_shape + event_shape`.
  
      Args:
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        variance: Floating-point `Tensor` with shape identical to
          `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.
      """
      with self._name_and_control_scope(name):
        try:
          return self._variance(**kwargs)
        except NotImplementedError:
          try:
            return tf.nest.map_structure(tf.square, self._stddev(**kwargs))
          except NotImplementedError:
            pass
          raise
  
    def _stddev(self, **kwargs):
      raise NotImplementedError('stddev is not implemented: {}'.format(
          type(self).__name__))
  
    def stddev(self, name='stddev', **kwargs):
      """Standard deviation.
  
      Standard deviation is defined as,
  
      ```none
      stddev = E[(X - E[X])**2]**0.5
      ```
  
      where `X` is the random variable associated with this distribution, `E`
      denotes expectation, and `stddev.shape = batch_shape + event_shape`.
  
      Args:
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        stddev: Floating-point `Tensor` with shape identical to
          `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.
      """
  
      with self._name_and_control_scope(name):
        try:
          return self._stddev(**kwargs)
        except NotImplementedError:
          try:
            return tf.nest.map_structure(tf.sqrt, self._variance(**kwargs))
          except NotImplementedError:
            pass
          raise
  
    def _covariance(self, **kwargs):
      raise NotImplementedError('covariance is not implemented: {}'.format(
          type(self).__name__))
  
    def covariance(self, name='covariance', **kwargs):
      """Covariance.
  
      Covariance is (possibly) defined only for non-scalar-event distributions.
  
      For example, for a length-`k`, vector-valued distribution, it is calculated
      as,
  
      ```none
      Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
      ```
  
      where `Cov` is a (batch of) `k x k` matrix, `0 <= (i, j) < k`, and `E`
      denotes expectation.
  
      Alternatively, for non-vector, multivariate distributions (e.g.,
      matrix-valued, Wishart), `Covariance` shall return a (batch of) matrices
      under some vectorization of the events, i.e.,
  
      ```none
      Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
      ```
  
      where `Cov` is a (batch of) `k' x k'` matrices,
      `0 <= (i, j) < k' = reduce_prod(event_shape)`, and `Vec` is some function
      mapping indices of this distribution's event dimensions to indices of a
      length-`k'` vector.
  
      Args:
        name: Python `str` prepended to names of ops created by this function.
        **kwargs: Named arguments forwarded to subclass implementation.
  
      Returns:
        covariance: Floating-point `Tensor` with shape `[B1, ..., Bn, k', k']`
          where the first `n` dimensions are batch coordinates and
          `k' = reduce_prod(self.event_shape)`.
      """
      with self._name_and_control_scope(name):
        return self._covariance(**kwargs)
  
    def _mode(self, **kwargs):
      raise NotImplementedError('mode is not implemented: {}'.format(
          type(self).__name__))
  
    def mode(self, name='mode', **kwargs):
      """Mode."""
      with self._name_and_control_scope(name):
        return self._mode(**kwargs)
  
    def _cross_entropy(self, other):
      return kullback_leibler.cross_entropy(
          self, other, allow_nan_stats=self.allow_nan_stats)
  
    def cross_entropy(self, other, name='cross_entropy'):
      """Computes the (Shannon) cross entropy.
  
      Denote this distribution (`self`) by `P` and the `other` distribution by
      `Q`. Assuming `P, Q` are absolutely continuous with respect to
      one another and permit densities `p(x) dr(x)` and `q(x) dr(x)`, (Shannon)
      cross entropy is defined as:
  
      ```none
      H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
      ```
  
      where `F` denotes the support of the random variable `X ~ P`.
  
      Args:
        other: `tfp.distributions.Distribution` instance.
        name: Python `str` prepended to names of ops created by this function.
  
      Returns:
        cross_entropy: `self.dtype` `Tensor` with shape `[B1, ..., Bn]`
          representing `n` different calculations of (Shannon) cross entropy.
      """
      with self._name_and_control_scope(name):
        return self._cross_entropy(other)
  
    def _kl_divergence(self, other):
      return kullback_leibler.kl_divergence(
          self, other, allow_nan_stats=self.allow_nan_stats)
  
    def kl_divergence(self, other, name='kl_divergence'):
      """Computes the Kullback--Leibler divergence.
  
      Denote this distribution (`self`) by `p` and the `other` distribution by
      `q`. Assuming `p, q` are absolutely continuous with respect to reference
      measure `r`, the KL divergence is defined as:
  
      ```none
      KL[p, q] = E_p[log(p(X)/q(X))]
               = -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
               = H[p, q] - H[p]
      ```
  
      where `F` denotes the support of the random variable `X ~ p`, `H[., .]`
      denotes (Shannon) cross entropy, and `H[.]` denotes (Shannon) entropy.
  
      Args:
        other: `tfp.distributions.Distribution` instance.
        name: Python `str` prepended to names of ops created by this function.
  
      Returns:
        kl_divergence: `self.dtype` `Tensor` with shape `[B1, ..., Bn]`
          representing `n` different calculations of the Kullback-Leibler
          divergence.
      """
      # NOTE: We do not enter a `self._name_and_control_scope` here.  We rely on
      # `tfd.kl_divergence(self, other)` to use `_name_and_control_scope` to apply
      # assertions on both Distributions.
      #
      # Subclasses that override `Distribution.kl_divergence` or `_kl_divergence`
      # must ensure that assertions are applied for both `self` and `other`.
      return self._kl_divergence(other)
  
    def _default_event_space_bijector(self, *args, **kwargs):
      raise NotImplementedError(
          '_default_event_space_bijector` is not implemented: {}'.format(
              type(self).__name__))
  
    @deprecation.deprecated(
        '2020-10-20',
        'Use `experimental_default_event_space_bijector` instead.')
    def _experimental_default_event_space_bijector(self, *args, **kwargs):
      return self.experimental_default_event_space_bijector(*args, **kwargs)
  
    def experimental_default_event_space_bijector(self, *args, **kwargs):
      """Bijector mapping the reals (R**n) to the event space of the distribution.
  
      Distributions with continuous support may implement
      `_default_event_space_bijector` which returns a subclass of
      `tfp.bijectors.Bijector` that maps R**n to the distribution's event space.
      For example, the default bijector for the `Beta` distribution
      is `tfp.bijectors.Sigmoid()`, which maps the real line to `[0, 1]`, the
      support of the `Beta` distribution. The default bijector for the
      `CholeskyLKJ` distribution is `tfp.bijectors.CorrelationCholesky`, which
      maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular
      matrices with ones along the diagonal.
  
      The purpose of `experimental_default_event_space_bijector` is
      to enable gradient descent in an unconstrained space for Variational
      Inference and Hamiltonian Monte Carlo methods. Some effort has been made to
      choose bijectors such that the tails of the distribution in the
      unconstrained space are between Gaussian and Exponential.
  
      For distributions with discrete event space, or for which TFP currently
      lacks a suitable bijector, this function returns `None`.
  
      Args:
        *args: Passed to implementation `_default_event_space_bijector`.
        **kwargs: Passed to implementation `_default_event_space_bijector`.
  
      Returns:
        event_space_bijector: `Bijector` instance or `None`.
      """
      return self._default_event_space_bijector(*args, **kwargs)
  
    def __str__(self):
      if self.batch_shape:
        maybe_batch_shape = ', batch_shape=' + _str_tensorshape(self.batch_shape)
      else:
        maybe_batch_shape = ''
      if self.event_shape:
        maybe_event_shape = ', event_shape=' + _str_tensorshape(self.event_shape)
      else:
        maybe_event_shape = ''
      if self.dtype is not None:
        maybe_dtype = ', dtype=' + _str_dtype(self.dtype)
      else:
        maybe_dtype = ''
      return ('tfp.distributions.{type_name}('
              '"{self_name}"'
              '{maybe_batch_shape}'
              '{maybe_event_shape}'
              '{maybe_dtype})'.format(
                  type_name=type(self).__name__,
                  self_name=self.name or '<unknown>',
                  maybe_batch_shape=maybe_batch_shape,
                  maybe_event_shape=maybe_event_shape,
                  maybe_dtype=maybe_dtype))
  
    def __repr__(self):
      return ('<tfp.distributions.{type_name} '
              '\'{self_name}\''
              ' batch_shape={batch_shape}'
              ' event_shape={event_shape}'
              ' dtype={dtype}>'.format(
                  type_name=type(self).__name__,
                  self_name=self.name or '<unknown>',
                  batch_shape=_str_tensorshape(self.batch_shape),
                  event_shape=_str_tensorshape(self.event_shape),
                  dtype=_str_dtype(self.dtype)))
  
    @contextlib.contextmanager
    def _name_and_control_scope(self, name=None, value=UNSET_VALUE, kwargs=None):
      """Helper function to standardize op scope."""
      # Note: we recieve `kwargs` and not `**kwargs` to ensure no collisions on
      # other args we choose to take in this function.
      with tf.name_scope(self.name):
        with tf.name_scope(name) as name_scope:
          deps = []
          if self._defer_all_assertions:
            deps.extend(self._parameter_control_dependencies(is_init=True))
          else:
            deps.extend(self._initial_parameter_control_dependencies)
          deps.extend(self._parameter_control_dependencies(is_init=False))
          if value is not UNSET_VALUE:
            deps.extend(self._sample_control_dependencies(
                value, **({} if kwargs is None else kwargs)))
          if not deps:
            yield name_scope
            return
          # In eager mode, some `assert_util.assert_xyz` calls return None. If a
          # Distribution is created in eager mode with `validate_args=True`, then
          # used in a `tf.function` context, it can result in errors when
          # `tf.convert_to_tensor` is called on the inputs to
          # `tf.control_dependencies` below. To avoid these errors, we drop the
          # `None`s here.
          deps = [x for x in deps if x is not None]
          with tf.control_dependencies(deps) as deps_scope:
            yield deps_scope
  
    def _expand_sample_shape_to_vector(self, x, name):
      """Helper to `sample` which ensures input is 1D."""
      prod = ps.reduce_prod(x)
      x = distribution_util.expand_to_vector(x, tensor_name=name)
      return x, prod
  
    def _set_sample_static_shape(self, x, sample_shape):
      """Helper to `sample`; sets static shape info."""
      batch_shape = self.batch_shape
      if (tf.nest.is_nested(self.dtype)
          and not tf.nest.is_nested(batch_shape)):
        batch_shape = tf.nest.map_structure(
            lambda _: batch_shape, self.dtype)
  
      return tf.nest.map_structure(
          functools.partial(
              _set_sample_static_shape_for_tensor, sample_shape=sample_shape),
          x, self.event_shape, batch_shape)
  
    def _is_scalar_helper(self, static_shape, dynamic_shape_fn):
      """Implementation for `is_scalar_batch` and `is_scalar_event`."""
      if tensorshape_util.rank(static_shape) is not None:
        return tensorshape_util.rank(static_shape) == 0
      shape = dynamic_shape_fn()
      if tf.compat.dimension_value(shape.shape[0]) is not None:
        # If the static_shape is correctly written then we should never execute
        # this branch. We keep it just in case there's some unimagined corner
        # case.
        return tensorshape_util.as_list(shape.shape) == [0]
      return tf.equal(tf.shape(shape)[0], 0)
  
    def _parameter_control_dependencies(self, is_init):
      """Returns a list of ops to be executed in members with graph deps.
  
      Typically subclasses override this function to return parameter specific
      assertions (eg, positivity of `scale`, etc.).
  
      Args:
        is_init: Python `bool` indicating that the call site is `__init__`.
  
      Returns:
        dependencies: `list`-like of ops to be executed in member functions with
          graph dependencies.
      """
      return ()
  
    def _sample_control_dependencies(self, value, **kwargs):
      """Returns a list of ops to be executed to validate distribution samples.
  
      The ops are executed in methods that take distribution samples as an
      argument (e.g. `log_prob` and `cdf`). They validate that `value` is
      within the support of the distribution. Typically subclasses override this
      function to return assertions specific to the distribution (e.g. samples
      from `Beta` must be between `0` and `1`). By convention, finite bounds of
      the support are considered valid samples, since `sample` may output values
      that are numerically equivalent to the bounds.
  
      Args:
        value: `float` or `double` `Tensor`.
        **kwargs: Additional keyword args.
  
      Returns:
        assertions: `list`-like of ops to be executed in member functions that
          take distribution samples as input.
      """
      return ()
  
  
  class _PrettyDict(dict):
    """`dict` with stable `repr`, `str`."""
  
    def __str__(self):
      pairs = (': '.join([str(k), str(v)]) for k, v in sorted(self.items()))
      return '{' + ', '.join(pairs) + '}'
  
    def __repr__(self):
      pairs = (': '.join([repr(k), repr(v)]) for k, v in sorted(self.items()))
      return '{' + ', '.join(pairs) + '}'
  
  
  def _recursively_replace_dict_for_pretty_dict(x):
    """Recursively replace `dict`s with `_PrettyDict`."""
    # We use "PrettyDict" because collections.OrderedDict repr/str has the word
    # "OrderedDict" in it. We only want to print "OrderedDict" if in fact the
    # input really is an OrderedDict.
    if isinstance(x, dict):
      return _PrettyDict({
          k: _recursively_replace_dict_for_pretty_dict(v)
          for k, v in x.items()})
    if (isinstance(x, collections.Sequence) and
        not isinstance(x, six.string_types)):
      args = (_recursively_replace_dict_for_pretty_dict(x_) for x_ in x)
      is_named_tuple = (isinstance(x, tuple) and
                        hasattr(x, '_asdict') and
                        hasattr(x, '_fields'))
      return type(x)(*args) if is_named_tuple else type(x)(args)
    if isinstance(x, collections.Mapping):
      return type(x)(**{k: _recursively_replace_dict_for_pretty_dict(v)
                        for k, v in x.items()})
    return x
  
  
  def _str_tensorshape(x):
    def _str(s):
      if tensorshape_util.rank(s) is None:
        return '?'
      return str(tensorshape_util.as_list(s)).replace('None', '?')
    # Because Python2 `dict`s are unordered, we must replace them with
    # `PrettyDict`s so __str__, __repr__ are deterministic.
    x = _recursively_replace_dict_for_pretty_dict(x)
    return str(tf.nest.map_structure(_str, x)).replace('\'', '')
  
  
  def _str_dtype(x):
    def _str(s):
      if s is None:
        return '?'
      return dtype_util.name(s)
    # Because Python2 `dict`s are unordered, we must replace them with
    # `PrettyDict`s so __str__, __repr__ are deterministic.
    x = _recursively_replace_dict_for_pretty_dict(x)
    return str(tf.nest.map_structure(_str, x)).replace('\'', '')
  
  
  def _get_base_shape_tensor(x):
    """Extracts an object's runtime (Tensor) shape for batch shape inference."""
    if hasattr(x, 'batch_shape_tensor'):  # `x` is a distribution.
      return x.batch_shape_tensor()
    elif hasattr(x, 'forward'):  # `x` is a bijector.
      # TODO(b/174778703): annotate batch shapes for bijectors.
      raise NotImplementedError('Bijector batch shapes are not implemented.')
    return ps.shape(x)
  
  
  def _get_base_shape(x):
    """Extracts an object's shape for batch shape inference."""
    if hasattr(x, 'batch_shape'):  # `x` is a distribution.
      return x.batch_shape
    elif hasattr(x, 'shape'):  # `x` is a Tensor or ndarray.
      return tf.TensorShape(x.shape)
    elif hasattr(x, 'forward'):  # `x` is a bijector.
      # TODO(b/174778703): annotate batch shapes for bijectors.
      return tf.TensorShape(None)
    # `x` is a Python list, tuple, or literal.
    return tf.TensorShape(np.array(x).shape)
  
  
  def _parameter_batch_shape_tensor(base_shape, event_ndims):
    base_shape = tf.convert_to_tensor(base_shape, dtype_hint=tf.int32)
    base_rank = ps.rank_from_shape(base_shape)
    return base_shape[:(base_rank -
                        # Don't try to slice away more ndims than the parameter
                        # actually has, if that's fewer than `event_ndims` (i.e.,
                        # if it relies on broadcasting).
                        ps.minimum(event_ndims, base_rank))]
  
  
  def _parameter_batch_shape(base_shape, event_ndims):
    if tf.is_tensor(event_ndims):
      event_ndims = tf.get_static_value(event_ndims)
      if event_ndims is None:
        return tf.TensorShape(None)
    return base_shape[:(len(base_shape) -
                        # Don't try to slice away more ndims than the parameter
                        # actually has, if that's fewer than `event_ndims` (i.e.,
                        # if it relies on broadcasting).
                        min(event_ndims, len(base_shape)))]
  