text: # Copyright 2016 The TensorFlow Authors. All Rights Reserved.
  #
  # 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.
  # ==============================================================================
  """Utilities for probability distributions."""
  
  from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  
  import functools
  import hashlib
  import numpy as np
  
  from tensorflow.python.framework import constant_op
  from tensorflow.python.framework import dtypes
  from tensorflow.python.framework import ops
  from tensorflow.python.framework import tensor_shape
  from tensorflow.python.framework import tensor_util
  from tensorflow.python.ops import array_ops
  from tensorflow.python.ops import check_ops
  from tensorflow.python.ops import control_flow_ops
  from tensorflow.python.ops import linalg_ops
  from tensorflow.python.ops import math_ops
  from tensorflow.python.ops import nn
  from tensorflow.python.util import tf_inspect
  
  
  def assert_close(
      x, y, data=None, summarize=None, message=None, name="assert_close"):
    """Assert that x and y are within machine epsilon of each other.
  
    Args:
      x: Floating-point `Tensor`
      y: Floating-point `Tensor`
      data: The tensors to print out if the condition is `False`. Defaults to
        error message and first few entries of `x` and `y`.
      summarize: Print this many entries of each tensor.
      message: A string to prefix to the default message.
      name: A name for this operation (optional).
  
    Returns:
      Op raising `InvalidArgumentError` if |x - y| > machine epsilon.
    """
    message = message or ""
    x = ops.convert_to_tensor(x, name="x")
    y = ops.convert_to_tensor(y, name="y")
  
    if data is None:
      data = [
          message,
          "Condition x ~= y did not hold element-wise: x = ", x, "y = ", y
      ]
  
    if x.dtype.is_integer:
      return check_ops.assert_equal(
          x, y, data=data, summarize=summarize, message=message, name=name)
  
    with ops.name_scope(name, "assert_close", [x, y, data]):
      tol = np.finfo(x.dtype.as_numpy_dtype).eps
      condition = math_ops.reduce_all(math_ops.less_equal(math_ops.abs(x-y), tol))
      return control_flow_ops.Assert(
          condition, data, summarize=summarize)
  
  
  def assert_integer_form(
      x, data=None, summarize=None, message=None,
      int_dtype=None, name="assert_integer_form"):
    """Assert that x has integer components (or floats equal to integers).
  
    Args:
      x: Floating-point `Tensor`
      data: The tensors to print out if the condition is `False`. Defaults to
        error message and first few entries of `x` and `y`.
      summarize: Print this many entries of each tensor.
      message: A string to prefix to the default message.
      int_dtype: A `tf.dtype` used to cast the float to. The default (`None`)
        implies the smallest possible signed int will be used for casting.
      name: A name for this operation (optional).
  
    Returns:
      Op raising `InvalidArgumentError` if `cast(x, int_dtype) != x`.
    """
    with ops.name_scope(name, values=[x, data]):
      x = ops.convert_to_tensor(x, name="x")
      if x.dtype.is_integer:
        return control_flow_ops.no_op()
      message = message or "{} has non-integer components".format(x)
      if int_dtype is None:
        try:
          int_dtype = {
              dtypes.float16: dtypes.int16,
              dtypes.float32: dtypes.int32,
              dtypes.float64: dtypes.int64,
          }[x.dtype.base_dtype]
        except KeyError:
          raise TypeError("Unrecognized type {}".format(x.dtype.name))
      return check_ops.assert_equal(
          x, math_ops.cast(math_ops.cast(x, int_dtype), x.dtype),
          data=data, summarize=summarize, message=message, name=name)
  
  
  def assert_symmetric(matrix):
    matrix_t = array_ops.matrix_transpose(matrix)
    return control_flow_ops.with_dependencies(
        [check_ops.assert_equal(matrix, matrix_t)], matrix)
  
  
  def embed_check_nonnegative_integer_form(
      x, name="embed_check_nonnegative_integer_form"):
    """Assert x is a non-negative tensor, and optionally of integers."""
    with ops.name_scope(name, values=[x]):
      x = ops.convert_to_tensor(x, name="x")
      assertions = [
          check_ops.assert_non_negative(
              x, message="'{}' must be non-negative.".format(x)),
      ]
      if not x.dtype.is_integer:
        assertions += [
            assert_integer_form(
                x, message="'{}' cannot contain fractional components.".format(
                    x)),
        ]
      return control_flow_ops.with_dependencies(assertions, x)
  
  
  def same_dynamic_shape(a, b):
    """Returns whether a and b have the same dynamic shape.
  
    Args:
      a: `Tensor`
      b: `Tensor`
  
    Returns:
      `bool` `Tensor` representing if both tensors have the same shape.
    """
    a = ops.convert_to_tensor(a, name="a")
    b = ops.convert_to_tensor(b, name="b")
  
    # Here we can't just do math_ops.equal(a.shape, b.shape), since
    # static shape inference may break the equality comparison between
    # shape(a) and shape(b) in math_ops.equal.
    def all_shapes_equal():
      return math_ops.reduce_all(math_ops.equal(
          array_ops.concat([array_ops.shape(a), array_ops.shape(b)], 0),
          array_ops.concat([array_ops.shape(b), array_ops.shape(a)], 0)))
  
    # One of the shapes isn't fully defined, so we need to use the dynamic
    # shape.
    return control_flow_ops.cond(
        math_ops.equal(array_ops.rank(a), array_ops.rank(b)),
        all_shapes_equal,
        lambda: constant_op.constant(False))
  
  
  def maybe_get_static_value(x, dtype=None):
    """Helper which tries to return a static value.
  
    Given `x`, extract it's value statically, optionally casting to a specific
    dtype. If this is not possible, None is returned.
  
    Args:
      x: `Tensor` for which to extract a value statically.
      dtype: Optional dtype to cast to.
  
    Returns:
      Statically inferred value if possible, otherwise None.
    """
    if x is None:
      return x
    try:
      # This returns an np.ndarray.
      x_ = tensor_util.constant_value(x)
    except TypeError:
      x_ = x
    if x_ is None or dtype is None:
      return x_
    return np.array(x_, dtype)
  
  
  def get_logits_and_probs(logits=None,
                           probs=None,
                           multidimensional=False,
                           validate_args=False,
                           name="get_logits_and_probs"):
    """Converts logit to probabilities (or vice-versa), and returns both.
  
    Args:
      logits: Floating-point `Tensor` representing log-odds.
      probs: Floating-point `Tensor` representing probabilities.
      multidimensional: Python `bool`, default `False`.
        If `True`, represents whether the last dimension of `logits` or `probs`,
        a `[N1, N2, ...  k]` dimensional tensor, representing the
        logit or probability of `shape[-1]` classes.
      validate_args: Python `bool`, default `False`. When `True`, either assert
        `0 <= probs <= 1` (if not `multidimensional`) or that the last dimension
        of `probs` sums to one.
      name: A name for this operation (optional).
  
    Returns:
      logits, probs: Tuple of `Tensor`s. If `probs` has an entry that is `0` or
        `1`, then the corresponding entry in the returned logit will be `-Inf` and
        `Inf` respectively.
  
    Raises:
      ValueError: if neither `probs` nor `logits` were passed in, or both were.
    """
    with ops.name_scope(name, values=[probs, logits]):
      if (probs is None) == (logits is None):
        raise ValueError("Must pass probs or logits, but not both.")
  
      if probs is None:
        logits = ops.convert_to_tensor(logits, name="logits")
        if not logits.dtype.is_floating:
          raise TypeError("logits must having floating type.")
        # We can early return since we constructed probs and therefore know
        # they're valid.
        if multidimensional:
          if validate_args:
            logits = embed_check_categorical_event_shape(logits)
          return logits, nn.softmax(logits, name="probs")
        return logits, math_ops.sigmoid(logits, name="probs")
  
      probs = ops.convert_to_tensor(probs, name="probs")
      if not probs.dtype.is_floating:
        raise TypeError("probs must having floating type.")
  
      if validate_args:
        with ops.name_scope("validate_probs"):
          one = constant_op.constant(1., probs.dtype)
          dependencies = [check_ops.assert_non_negative(probs)]
          if multidimensional:
            probs = embed_check_categorical_event_shape(probs)
            dependencies += [assert_close(math_ops.reduce_sum(probs, -1), one,
                                          message="probs does not sum to 1.")]
          else:
            dependencies += [check_ops.assert_less_equal(
                probs, one, message="probs has components greater than 1.")]
          probs = control_flow_ops.with_dependencies(dependencies, probs)
  
      with ops.name_scope("logits"):
        if multidimensional:
          # Here we don't compute the multidimensional case, in a manner
          # consistent with respect to the unidimensional case. We do so
          # following the TF convention. Typically, you might expect to see
          # logits = log(probs) - log(probs[pivot]). A side-effect of
          # being consistent with the TF approach is that the unidimensional case
          # implicitly handles the second dimension but the multidimensional case
          # explicitly keeps the pivot dimension.
          return math_ops.log(probs), probs
        return math_ops.log(probs) - math_ops.log1p(-1. * probs), probs
  
  
  def _is_known_unsigned_by_dtype(dt):
    """Helper returning True if dtype is known to be unsigned."""
    return {
        dtypes.bool: True,
        dtypes.uint8: True,
        dtypes.uint16: True,
    }.get(dt.base_dtype, False)
  
  
  def _is_known_signed_by_dtype(dt):
    """Helper returning True if dtype is known to be signed."""
    return {
        dtypes.float16: True,
        dtypes.float32: True,
        dtypes.float64: True,
        dtypes.int8: True,
        dtypes.int16: True,
        dtypes.int32: True,
        dtypes.int64: True,
    }.get(dt.base_dtype, False)
  
  
  def _is_known_dtype(dt):
    """Helper returning True if dtype is known."""
    return _is_known_unsigned_by_dtype(dt) or _is_known_signed_by_dtype(dt)
  
  
  def _largest_integer_by_dtype(dt):
    """Helper returning the largest integer exactly representable by dtype."""
    if not _is_known_dtype(dt):
      raise TypeError("Unrecognized dtype: {}".format(dt.name))
    if dt.is_floating:
      return int(2**(np.finfo(dt.as_numpy_dtype).nmant + 1))
    if dt.is_integer:
      return np.iinfo(dt.as_numpy_dtype).max
    if dt.base_dtype == dtypes.bool:
      return int(1)
    # We actually can't land here but keep the case for completeness.
    raise TypeError("Unrecognized dtype: {}".format(dt.name))
  
  
  def _smallest_integer_by_dtype(dt):
    """Helper returning the smallest integer exactly representable by dtype."""
    if not _is_known_dtype(dt):
      raise TypeError("Unrecognized dtype: {}".format(dt.name))
    if _is_known_unsigned_by_dtype(dt):
      return 0
    return -1 * _largest_integer_by_dtype(dt)
  
  
  def _is_integer_like_by_dtype(dt):
    """Helper returning True if dtype.is_integer or is `bool`."""
    if not _is_known_dtype(dt):
      raise TypeError("Unrecognized dtype: {}".format(dt.name))
    return dt.is_integer or dt.base_dtype == dtypes.bool
  
  
  def embed_check_categorical_event_shape(
      categorical_param,
      name="embed_check_categorical_event_shape"):
    """Embeds checks that categorical distributions don't have too many classes.
  
    A categorical-type distribution is one which, e.g., returns the class label
    rather than a one-hot encoding.  E.g., `Categorical(probs)`.
  
    Since distributions output samples in the same dtype as the parameters, we
    must ensure that casting doesn't lose precision. That is, the
    `parameter.dtype` implies a maximum number of classes. However, since shape is
    `int32` and categorical variables are presumed to be indexes into a `Tensor`,
    we must also ensure that the number of classes is no larger than the largest
    possible `int32` index, i.e., `2**31-1`.
  
    In other words the number of classes, `K`, must satisfy the following
    condition:
  
    ```python
    K <= min(
        int(2**31 - 1),  # Largest float as an index.
        {
            dtypes.float16: int(2**11),   # Largest int as a float16.
            dtypes.float32: int(2**24),
            dtypes.float64: int(2**53),
        }.get(categorical_param.dtype.base_dtype, 0))
    ```
  
    Args:
      categorical_param: Floating-point `Tensor` representing parameters of
        distribution over categories. The rightmost shape is presumed to be the
        number of categories.
      name: A name for this operation (optional).
  
    Returns:
      categorical_param: Input `Tensor` with appropriate assertions embedded.
  
    Raises:
      TypeError: if `categorical_param` has an unknown `dtype`.
      ValueError: if we can statically identify `categorical_param` as being too
        large (for being closed under int32/float casting).
    """
    with ops.name_scope(name, values=[categorical_param]):
      x = ops.convert_to_tensor(categorical_param, name="categorical_param")
      # The size must not exceed both of:
      # - The largest possible int32 (since categorical values are presumed to be
      #   indexes into a Tensor).
      # - The largest possible integer exactly representable under the given
      #   floating-point dtype (since we need to cast to/from).
      #
      # The chosen floating-point thresholds are 2**(1 + mantissa_bits).
      # For more details, see:
      # https://en.wikipedia.org/wiki/Floating-point_arithmetic#Internal_representation
      x_dtype = x.dtype.base_dtype
      max_event_size = (_largest_integer_by_dtype(x_dtype)
                        if x_dtype.is_floating else 0)
      if max_event_size is 0:
        raise TypeError("Unable to validate size of unrecognized dtype "
                        "({}).".format(x_dtype.name))
      try:
        x_shape_static = x.get_shape().with_rank_at_least(1)
      except ValueError:
        raise ValueError("A categorical-distribution parameter must have "
                         "at least 1 dimension.")
      if x_shape_static[-1].value is not None:
        event_size = x_shape_static[-1].value
        if event_size < 2:
          raise ValueError("A categorical-distribution parameter must have at "
                           "least 2 events.")
        if event_size > max_event_size:
          raise ValueError(
              "Number of classes exceeds `dtype` precision, i.e., "
              "{} implies shape ({}) cannot exceed {}.".format(
                  x_dtype.name, event_size, max_event_size))
        return x
      else:
        event_size = array_ops.shape(x, name="x_shape")[-1]
        return control_flow_ops.with_dependencies([
            check_ops.assert_rank_at_least(
                x, 1, message=("A categorical-distribution parameter must have "
                               "at least 1 dimension.")),
            check_ops.assert_greater_equal(
                array_ops.shape(x)[-1], 2,
                message=("A categorical-distribution parameter must have at "
                         "least 2 events.")),
            check_ops.assert_less_equal(
                event_size, max_event_size,
                message="Number of classes exceeds `dtype` precision, "
                        "i.e., {} dtype cannot exceed {} shape.".format(
                            x_dtype.name, max_event_size)),
        ], x)
  
  
  def embed_check_integer_casting_closed(
      x,
      target_dtype,
      assert_nonnegative=True,
      name="embed_check_casting_closed"):
    """Ensures integers remain unaffected despite casting to/from int/float types.
  
    Example integer-types: `uint8`, `int32`, `bool`.
    Example floating-types: `float32`, `float64`.
  
    The largest possible integer representable by an IEEE754 floating-point is
    `2**(1 + mantissa_bits)` yet the largest possible integer as an int-type is
    `2**(bits - 1) - 1`. This function ensures that a `Tensor` purporting to have
    integer-form values can be cast to some other type without loss of precision.
  
    The smallest representable integer is the negative of the largest
    representable integer, except for types: `uint8`, `uint16`, `bool`. For these
    types, the smallest representable integer is `0`.
  
    Args:
      x: `Tensor` representing integer-form values.
      target_dtype: TF `dtype` under which `x` should have identical values.
      assert_nonnegative: `bool` indicating `x` should contain nonnegative values.
      name: A name for this operation (optional).
  
    Returns:
      x: Input `Tensor` with appropriate assertions embedded.
  
    Raises:
      TypeError: if `x` is neither integer- nor floating-type.
      TypeError: if `target_dtype` is neither integer- nor floating-type.
      TypeError: if neither `x` nor `target_dtype` are integer-type.
    """
  
    with ops.name_scope(name, values=[x]):
      x = ops.convert_to_tensor(x, name="x")
      if (not _is_integer_like_by_dtype(x.dtype)
          and not x.dtype.is_floating):
        raise TypeError("{}.dtype must be floating- or "
                        "integer-type.".format(x.dtype.name))
      if (not _is_integer_like_by_dtype(target_dtype)
          and not target_dtype.is_floating):
        raise TypeError("target_dtype ({}) must be floating- or "
                        "integer-type.".format(target_dtype.name))
      if (not _is_integer_like_by_dtype(x.dtype)
          and not _is_integer_like_by_dtype(target_dtype)):
        raise TypeError("At least one of {}.dtype ({}) and target_dtype ({}) "
                        "must be integer-type.".format(
                            x, x.dtype.name, target_dtype.name))
  
      assertions = []
      if assert_nonnegative:
        assertions += [
            check_ops.assert_non_negative(
                x, message="Elements must be non-negative."),
        ]
  
      if x.dtype.is_floating:
        # Being here means _is_integer_like_by_dtype(target_dtype) = True.
        # Since this check implies the magnitude check below, we need only it.
        assertions += [
            assert_integer_form(
                x, int_dtype=target_dtype,
                message="Elements must be {}-equivalent.".format(
                    target_dtype.name)),
        ]
      else:
        if (_largest_integer_by_dtype(x.dtype)
            > _largest_integer_by_dtype(target_dtype)):
          # Cast may lose integer precision.
          assertions += [
              check_ops.assert_less_equal(
                  x, _largest_integer_by_dtype(target_dtype),
                  message=("Elements cannot exceed {}.".format(
                      _largest_integer_by_dtype(target_dtype)))),
          ]
        if (not assert_nonnegative and
            (_smallest_integer_by_dtype(x.dtype)
             < _smallest_integer_by_dtype(target_dtype))):
          assertions += [
              check_ops.assert_greater_equal(
                  x, _smallest_integer_by_dtype(target_dtype),
                  message=("Elements cannot be smaller than {}.".format(
                      _smallest_integer_by_dtype(target_dtype)))),
          ]
  
      if not assertions:
        return x
      return control_flow_ops.with_dependencies(assertions, x)
  
  
  def log_combinations(n, counts, name="log_combinations"):
    """Multinomial coefficient.
  
    Given `n` and `counts`, where `counts` has last dimension `k`, we compute
    the multinomial coefficient as:
  
    ```n! / sum_i n_i!```
  
    where `i` runs over all `k` classes.
  
    Args:
      n: Floating-point `Tensor` broadcastable with `counts`. This represents `n`
        outcomes.
      counts: Floating-point `Tensor` broadcastable with `n`. This represents
        counts in `k` classes, where `k` is the last dimension of the tensor.
      name: A name for this operation (optional).
  
    Returns:
      `Tensor` representing the multinomial coefficient between `n` and `counts`.
    """
    # First a bit about the number of ways counts could have come in:
    # E.g. if counts = [1, 2], then this is 3 choose 2.
    # In general, this is (sum counts)! / sum(counts!)
    # The sum should be along the last dimension of counts. This is the
    # "distribution" dimension. Here n a priori represents the sum of counts.
    with ops.name_scope(name, values=[n, counts]):
      n = ops.convert_to_tensor(n, name="n")
      counts = ops.convert_to_tensor(counts, name="counts")
      total_permutations = math_ops.lgamma(n + 1)
      counts_factorial = math_ops.lgamma(counts + 1)
      redundant_permutations = math_ops.reduce_sum(counts_factorial, axis=[-1])
      return total_permutations - redundant_permutations
  
  
  def matrix_diag_transform(matrix, transform=None, name=None):
    """Transform diagonal of [batch-]matrix, leave rest of matrix unchanged.
  
    Create a trainable covariance defined by a Cholesky factor:
  
    ```python
    # Transform network layer into 2 x 2 array.
    matrix_values = tf.contrib.layers.fully_connected(activations, 4)
    matrix = tf.reshape(matrix_values, (batch_size, 2, 2))
  
    # Make the diagonal positive. If the upper triangle was zero, this would be a
    # valid Cholesky factor.
    chol = matrix_diag_transform(matrix, transform=tf.nn.softplus)
  
    # LinearOperatorLowerTriangular ignores the upper triangle.
    operator = LinearOperatorLowerTriangular(chol)
    ```
  
    Example of heteroskedastic 2-D linear regression.
  
    ```python
    # Get a trainable Cholesky factor.
    matrix_values = tf.contrib.layers.fully_connected(activations, 4)
    matrix = tf.reshape(matrix_values, (batch_size, 2, 2))
    chol = matrix_diag_transform(matrix, transform=tf.nn.softplus)
  
    # Get a trainable mean.
    mu = tf.contrib.layers.fully_connected(activations, 2)
  
    # This is a fully trainable multivariate normal!
    dist = tf.contrib.distributions.MVNCholesky(mu, chol)
  
    # Standard log loss. Minimizing this will "train" mu and chol, and then dist
    # will be a distribution predicting labels as multivariate Gaussians.
    loss = -1 * tf.reduce_mean(dist.log_prob(labels))
    ```
  
    Args:
      matrix:  Rank `R` `Tensor`, `R >= 2`, where the last two dimensions are
        equal.
      transform:  Element-wise function mapping `Tensors` to `Tensors`. To
        be applied to the diagonal of `matrix`. If `None`, `matrix` is returned
        unchanged. Defaults to `None`.
      name:  A name to give created ops.
        Defaults to "matrix_diag_transform".
  
    Returns:
      A `Tensor` with same shape and `dtype` as `matrix`.
    """
    with ops.name_scope(name, "matrix_diag_transform", [matrix]):
      matrix = ops.convert_to_tensor(matrix, name="matrix")
      if transform is None:
        return matrix
      # Replace the diag with transformed diag.
      diag = array_ops.matrix_diag_part(matrix)
      transformed_diag = transform(diag)
      transformed_mat = array_ops.matrix_set_diag(matrix, transformed_diag)
  
    return transformed_mat
  
  
  def rotate_transpose(x, shift, name="rotate_transpose"):
    """Circularly moves dims left or right.
  
    Effectively identical to:
  
    ```python
    numpy.transpose(x, numpy.roll(numpy.arange(len(x.shape)), shift))
    ```
  
    When `validate_args=False` additional graph-runtime checks are
    performed. These checks entail moving data from to GPU to CPU.
  
    Example:
  
    ```python
    x = tf.random_normal([1, 2, 3, 4])  # Tensor of shape [1, 2, 3, 4].
    rotate_transpose(x, -1).shape == [2, 3, 4, 1]
    rotate_transpose(x, -2).shape == [3, 4, 1, 2]
    rotate_transpose(x,  1).shape == [4, 1, 2, 3]
    rotate_transpose(x,  2).shape == [3, 4, 1, 2]
    rotate_transpose(x,  7).shape == rotate_transpose(x, 3).shape  # [2, 3, 4, 1]
    rotate_transpose(x, -7).shape == rotate_transpose(x, -3).shape  # [4, 1, 2, 3]
    ```
  
    Args:
      x: `Tensor`.
      shift: `Tensor`. Number of dimensions to transpose left (shift<0) or
        transpose right (shift>0).
      name: Python `str`. The name to give this op.
  
    Returns:
      rotated_x: Input `Tensor` with dimensions circularly rotated by shift.
  
    Raises:
      TypeError: if shift is not integer type.
    """
    with ops.name_scope(name, values=[x, shift]):
      x = ops.convert_to_tensor(x, name="x")
      shift = ops.convert_to_tensor(shift, name="shift")
      # We do not assign back to preserve constant-ness.
      check_ops.assert_integer(shift)
      shift_value_static = tensor_util.constant_value(shift)
      ndims = x.get_shape().ndims
      if ndims is not None and shift_value_static is not None:
        if ndims < 2: return x
        shift_value_static = np.sign(shift_value_static) * (
            abs(shift_value_static) % ndims)
        if shift_value_static == 0: return x
        perm = np.roll(np.arange(ndims), shift_value_static)
        return array_ops.transpose(x, perm=perm)
      else:
        # Consider if we always had a positive shift, and some specified
        # direction.
        # When shifting left we want the new array:
        #   last(x, n-shift) + first(x, shift)
        # and if shifting right then we want:
        #   last(x, shift) + first(x, n-shift)
        # Observe that last(a) == slice(a, n) and first(a) == slice(0, a).
        # Also, we can encode direction and shift as one: direction * shift.
        # Combining these facts, we have:
        #   a = cond(shift<0, -shift, n-shift)
        #   last(x, n-a) + first(x, a) == x[a:n] + x[0:a]
        # Finally, we transform shift by modulo length so it can be specified
        # independently from the array upon which it operates (like python).
        ndims = array_ops.rank(x)
        shift = array_ops.where(math_ops.less(shift, 0),
                                math_ops.mod(-shift, ndims),
                                ndims - math_ops.mod(shift, ndims))
        first = math_ops.range(0, shift)
        last = math_ops.range(shift, ndims)
        perm = array_ops.concat([last, first], 0)
        return array_ops.transpose(x, perm=perm)
  
  
  def pick_vector(cond,
                  true_vector,
                  false_vector,
                  name="pick_vector"):
    """Picks possibly different length row `Tensor`s based on condition.
  
    Value `Tensor`s should have exactly one dimension.
  
    If `cond` is a python Boolean or `tf.constant` then either `true_vector` or
    `false_vector` is immediately returned. I.e., no graph nodes are created and
    no validation happens.
  
    Args:
      cond: `Tensor`. Must have `dtype=tf.bool` and be scalar.
      true_vector: `Tensor` of one dimension. Returned when cond is `True`.
      false_vector: `Tensor` of one dimension. Returned when cond is `False`.
      name: Python `str`. The name to give this op.
  
    Example:
  
    ```python
    pick_vector(tf.less(0, 5), tf.range(10, 12), tf.range(15, 18))  # [10, 11]
    pick_vector(tf.less(5, 0), tf.range(10, 12), tf.range(15, 18))  # [15, 16, 17]
    ```
  
    Returns:
      true_or_false_vector: `Tensor`.
  
    Raises:
      TypeError: if `cond.dtype != tf.bool`
      TypeError: if `cond` is not a constant and
        `true_vector.dtype != false_vector.dtype`
    """
    with ops.name_scope(name, values=(cond, true_vector, false_vector)):
      cond = ops.convert_to_tensor(cond, name="cond")
      if cond.dtype != dtypes.bool:
        raise TypeError("%s.dtype=%s which is not %s" %
                        (cond, cond.dtype, dtypes.bool))
      cond_value_static = tensor_util.constant_value(cond)
      if cond_value_static is not None:
        return true_vector if cond_value_static else false_vector
      true_vector = ops.convert_to_tensor(true_vector, name="true_vector")
      false_vector = ops.convert_to_tensor(false_vector, name="false_vector")
      if true_vector.dtype != false_vector.dtype:
        raise TypeError(
            "%s.dtype=%s does not match %s.dtype=%s"
            % (true_vector, true_vector.dtype,
               false_vector, false_vector.dtype))
      n = array_ops.shape(true_vector)[0]
      return array_ops.slice(
          array_ops.concat([true_vector, false_vector], 0),
          [array_ops.where(cond, 0, n)], [array_ops.where(cond, n, -1)])
  
  
  def prefer_static_broadcast_shape(
      shape1, shape2, name="prefer_static_broadcast_shape"):
    """Convenience function which statically broadcasts shape when possible.
  
    Args:
      shape1:  `1-D` integer `Tensor`.  Already converted to tensor!
      shape2:  `1-D` integer `Tensor`.  Already converted to tensor!
      name:  A string name to prepend to created ops.
  
    Returns:
      The broadcast shape, either as `TensorShape` (if broadcast can be done
        statically), or as a `Tensor`.
    """
    with ops.name_scope(name, values=[shape1, shape2]):
      def make_shape_tensor(x):
        return ops.convert_to_tensor(x, name="shape", dtype=dtypes.int32)
  
      def get_tensor_shape(s):
        if isinstance(s, tensor_shape.TensorShape):
          return s
        s_ = tensor_util.constant_value(make_shape_tensor(s))
        if s_ is not None:
          return tensor_shape.TensorShape(s_)
        return None
  
      def get_shape_tensor(s):
        if not isinstance(s, tensor_shape.TensorShape):
          return make_shape_tensor(s)
        if s.is_fully_defined():
          return make_shape_tensor(s.as_list())
        raise ValueError("Cannot broadcast from partially "
                         "defined `TensorShape`.")
  
      shape1_ = get_tensor_shape(shape1)
      shape2_ = get_tensor_shape(shape2)
      if shape1_ is not None and shape2_ is not None:
        return array_ops.broadcast_static_shape(shape1_, shape2_)
  
      shape1_ = get_shape_tensor(shape1)
      shape2_ = get_shape_tensor(shape2)
      return array_ops.broadcast_dynamic_shape(shape1_, shape2_)
  
  
  def prefer_static_rank(x):
    """Return static rank of tensor `x` if available, else `tf.rank(x)`.
  
    Args:
      x: `Tensor` (already converted).
  
    Returns:
      Numpy array (if static rank is obtainable), else `Tensor`.
    """
    return prefer_static_value(array_ops.rank(x))
  
  
  def prefer_static_shape(x):
    """Return static shape of tensor `x` if available, else `tf.shape(x)`.
  
    Args:
      x: `Tensor` (already converted).
  
    Returns:
      Numpy array (if static shape is obtainable), else `Tensor`.
    """
    return prefer_static_value(array_ops.shape(x))
  
  
  def prefer_static_value(x):
    """Return static value of tensor `x` if available, else `x`.
  
    Args:
      x: `Tensor` (already converted).
  
    Returns:
      Numpy array (if static value is obtainable), else `Tensor`.
    """
    static_x = tensor_util.constant_value(x)
    if static_x is not None:
      return static_x
    return x
  
  
  def gen_new_seed(seed, salt):
    """Generate a new seed, from the given seed and salt."""
    if seed is None:
      return None
    string = (str(seed) + salt).encode("utf-8")
    return int(hashlib.md5(string).hexdigest()[:8], 16) & 0x7FFFFFFF
  
  
  def fill_triangular(x, upper=False, name=None):
    """Creates a (batch of) triangular matrix from a vector of inputs.
  
    Created matrix can be lower- or upper-triangular. (It is more efficient to
    create the matrix as upper or lower, rather than transpose.)
  
    Triangular matrix elements are filled in a clockwise spiral. See example,
    below.
  
    If `x.get_shape()` is `[b1, b2, ..., bB, d]` then the output shape is
    `[b1, b2, ..., bB, n, n]` where `n` is such that `d = n(n+1)/2`, i.e.,
    `n = int(np.sqrt(0.25 + 2. * m) - 0.5)`.
  
    Example:
  
    ```python
    fill_triangular([1, 2, 3, 4, 5, 6])
    # ==> [[4, 0, 0],
    #      [6, 5, 0],
    #      [3, 2, 1]]
  
    fill_triangular([1, 2, 3, 4, 5, 6], upper=True)
    # ==> [[1, 2, 3],
    #      [0, 5, 6],
    #      [0, 0, 4]]
    ```
  
    For comparison, a pure numpy version of this function can be found in
    `util_test.py`, function `_fill_triangular`.
  
    Args:
      x: `Tensor` representing lower (or upper) triangular elements.
      upper: Python `bool` representing whether output matrix should be upper
        triangular (`True`) or lower triangular (`False`, default).
      name: Python `str`. The name to give this op.
  
    Returns:
      tril: `Tensor` with lower (or upper) triangular elements filled from `x`.
  
    Raises:
      ValueError: if `x` cannot be mapped to a triangular matrix.
    """
  
    with ops.name_scope(name, "fill_triangular", values=[x]):
      x = ops.convert_to_tensor(x, name="x")
      if x.shape.with_rank_at_least(1)[-1].value is not None:
        # Formula derived by solving for n: m = n(n+1)/2.
        m = np.int32(x.shape[-1].value)
        n = np.sqrt(0.25 + 2. * m) - 0.5
        if n != np.floor(n):
          raise ValueError("Input right-most shape ({}) does not "
                           "correspond to a triangular matrix.".format(m))
        n = np.int32(n)
        static_final_shape = x.shape[:-1].concatenate([n, n])
      else:
        m = array_ops.shape(x)[-1]
        # For derivation, see above. Casting automatically lops off the 0.5, so we
        # omit it.  We don't validate n is an integer because this has
        # graph-execution cost; an error will be thrown from the reshape, below.
        n = math_ops.cast(
            math_ops.sqrt(0.25 + math_ops.cast(2 * m, dtype=dtypes.float32)),
            dtype=dtypes.int32)
        static_final_shape = x.shape.with_rank_at_least(1)[:-1].concatenate(
            [None, None])
      # We now concatenate the "tail" of `x` to `x` (and reverse one of them).
      #
      # We do this based on the insight that the input `x` provides `ceil(n/2)`
      # rows of an `n x n` matrix, some of which will get zeroed out being on the
      # wrong side of the diagonal. The first row will not get zeroed out at all,
      # and we need `floor(n/2)` more rows, so the first is what we omit from
      # `x_tail`. If we then stack those `ceil(n/2)` rows with the `floor(n/2)`
      # rows provided by a reversed tail, it is exactly the other set of elements
      # of the reversed tail which will be zeroed out for being on the wrong side
      # of the diagonal further up/down the matrix. And, in doing-so, we've filled
      # the triangular matrix in a clock-wise spiral pattern. Neat!
      #
      # Try it out in numpy:
      #  n = 3
      #  x = np.arange(n * (n + 1) / 2)
      #  m = x.shape[0]
      #  n = np.int32(np.sqrt(.25 + 2 * m) - .5)
      #  x_tail = x[(m - (n**2 - m)):]
      #  np.concatenate([x_tail, x[::-1]], 0).reshape(n, n)  # lower
      #  # ==> array([[3, 4, 5],
      #               [5, 4, 3],
      #               [2, 1, 0]])
      #  np.concatenate([x, x_tail[::-1]], 0).reshape(n, n)  # upper
      #  # ==> array([[0, 1, 2],
      #               [3, 4, 5],
      #               [5, 4, 3]])
      #
      # Note that we can't simply do `x[..., -(n**2 - m):]` because this doesn't
      # correctly handle `m == n == 1`. Hence, we do nonnegative indexing.
      # Furthermore observe that:
      #   m - (n**2 - m)
      #   = n**2 / 2 + n / 2 - (n**2 - n**2 / 2 + n / 2)
      #   = 2 (n**2 / 2 + n / 2) - n**2
      #   = n**2 + n - n**2
      #   = n
      ndims = prefer_static_rank(x)
      if upper:
        x_list = [x, array_ops.reverse(x[..., n:], axis=[ndims - 1])]
      else:
        x_list = [x[..., n:], array_ops.reverse(x, axis=[ndims - 1])]
      new_shape = (
          static_final_shape.as_list()
          if static_final_shape.is_fully_defined()
          else array_ops.concat([array_ops.shape(x)[:-1], [n, n]], axis=0))
      x = array_ops.reshape(array_ops.concat(x_list, axis=-1), new_shape)
      x = array_ops.matrix_band_part(
          x,
          num_lower=(0 if upper else -1),
          num_upper=(-1 if upper else 0))
      x.set_shape(static_final_shape)
      return x
  
  
  def fill_triangular_inverse(x, upper=False, name=None):
    """Creates a vector from a (batch of) triangular matrix.
  
    The vector is created from the lower-triangular or upper-triangular portion
    depending on the value of the parameter `upper`.
  
    If `x.shape` is `[b1, b2, ..., bB, n, n]` then the output shape is
    `[b1, b2, ..., bB, d]` where `d = n (n + 1) / 2`.
  
    Example:
  
    ```python
    fill_triangular_inverse(
      [[4, 0, 0],
       [6, 5, 0],
       [3, 2, 1]])
  
    # ==> [1, 2, 3, 4, 5, 6]
  
    fill_triangular_inverse(
      [[1, 2, 3],
       [0, 5, 6],
       [0, 0, 4]], upper=True)
  
    # ==> [1, 2, 3, 4, 5, 6]
    ```
  
    Args:
      x: `Tensor` representing lower (or upper) triangular elements.
      upper: Python `bool` representing whether output matrix should be upper
        triangular (`True`) or lower triangular (`False`, default).
      name: Python `str`. The name to give this op.
  
    Returns:
      flat_tril: (Batch of) vector-shaped `Tensor` representing vectorized lower
        (or upper) triangular elements from `x`.
    """
  
    with ops.name_scope(name, "fill_triangular_inverse", values=[x]):
      x = ops.convert_to_tensor(x, name="x")
      if x.shape.with_rank_at_least(2)[-1].value is not None:
        n = np.int32(x.shape[-1].value)
        m = np.int32((n * (n + 1)) // 2)
        static_final_shape = x.shape[:-2].concatenate([m])
      else:
        n = array_ops.shape(x)[-1]
        m = (n * (n + 1)) // 2
        static_final_shape = x.shape.with_rank_at_least(2)[:-2].concatenate(
            [None])
      ndims = prefer_static_rank(x)
      if upper:
        initial_elements = x[..., 0, :]
        triangular_portion = x[..., 1:, :]
      else:
        initial_elements = array_ops.reverse(x[..., -1, :], axis=[ndims - 2])
        triangular_portion = x[..., :-1, :]
      rotated_triangular_portion = array_ops.reverse(
          array_ops.reverse(triangular_portion, axis=[ndims - 1]),
          axis=[ndims - 2])
      consolidated_matrix = triangular_portion + rotated_triangular_portion
      end_sequence = array_ops.reshape(
          consolidated_matrix,
          array_ops.concat([array_ops.shape(x)[:-2], [n * (n - 1)]], axis=0))
      y = array_ops.concat([initial_elements, end_sequence[..., :m - n]], axis=-1)
      y.set_shape(static_final_shape)
      return y
  
  
  def tridiag(below=None, diag=None, above=None, name=None):
    """Creates a matrix with values set above, below, and on the diagonal.
  
    Example:
  
    ```python
    tridiag(below=[1., 2., 3.],
            diag=[4., 5., 6., 7.],
            above=[8., 9., 10.])
    # ==> array([[  4.,   8.,   0.,   0.],
    #            [  1.,   5.,   9.,   0.],
    #            [  0.,   2.,   6.,  10.],
    #            [  0.,   0.,   3.,   7.]], dtype=float32)
    ```
  
    Warning: This Op is intended for convenience, not efficiency.
  
    Args:
      below: `Tensor` of shape `[B1, ..., Bb, d-1]` corresponding to the below
        diagonal part. `None` is logically equivalent to `below = 0`.
      diag: `Tensor` of shape `[B1, ..., Bb, d]` corresponding to the diagonal
        part.  `None` is logically equivalent to `diag = 0`.
      above: `Tensor` of shape `[B1, ..., Bb, d-1]` corresponding to the above
        diagonal part.  `None` is logically equivalent to `above = 0`.
      name: Python `str`. The name to give this op.
  
    Returns:
      tridiag: `Tensor` with values set above, below and on the diagonal.
  
    Raises:
      ValueError: if all inputs are `None`.
    """
  
    def _pad(x):
      """Prepends and appends a zero to every vector in a batch of vectors."""
      shape = array_ops.concat([array_ops.shape(x)[:-1], [1]], axis=0)
      z = array_ops.zeros(shape, dtype=x.dtype)
      return array_ops.concat([z, x, z], axis=-1)
  
    def _add(*x):
      """Adds list of Tensors, ignoring `None`."""
      s = None
      for y in x:
        if y is None:
          continue
        elif s is None:
          s = y
        else:
          s += y
      if s is None:
        raise ValueError("Must specify at least one of `below`, `diag`, `above`.")
      return s
  
    with ops.name_scope(name, "tridiag", [below, diag, above]):
      if below is not None:
        below = ops.convert_to_tensor(below, name="below")
        below = array_ops.matrix_diag(_pad(below))[..., :-1, 1:]
      if diag is not None:
        diag = ops.convert_to_tensor(diag, name="diag")
        diag = array_ops.matrix_diag(diag)
      if above is not None:
        above = ops.convert_to_tensor(above, name="above")
        above = array_ops.matrix_diag(_pad(above))[..., 1:, :-1]
      # TODO(jvdillon): Consider using scatter_nd instead of creating three full
      # matrices.
      return _add(below, diag, above)
  
  
  def reduce_weighted_logsumexp(
      logx,
      w=None,
      axis=None,
      keep_dims=False,
      return_sign=False,
      name=None):
    """Computes `log(abs(sum(weight * exp(elements across tensor dimensions))))`.
  
    If all weights `w` are known to be positive, it is more efficient to directly
    use `reduce_logsumexp`, i.e., `tf.reduce_logsumexp(logx + tf.log(w))` is more
    efficient than `du.reduce_weighted_logsumexp(logx, w)`.
  
    Reduces `input_tensor` along the dimensions given in `axis`.
    Unless `keep_dims` is true, the rank of the tensor is reduced by 1 for each
    entry in `axis`. If `keep_dims` is true, the reduced dimensions
    are retained with length 1.
  
    If `axis` has no entries, all dimensions are reduced, and a
    tensor with a single element is returned.
  
    This function is more numerically stable than log(sum(w * exp(input))). It
    avoids overflows caused by taking the exp of large inputs and underflows
    caused by taking the log of small inputs.
  
    For example:
  
    ```python
    x = tf.constant([[0., 0, 0],
                     [0, 0, 0]])
  
    w = tf.constant([[-1., 1, 1],
                     [1, 1, 1]])
  
    du.reduce_weighted_logsumexp(x, w)
    # ==> log(-1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1) = log(4)
  
    du.reduce_weighted_logsumexp(x, w, axis=0)
    # ==> [log(-1+1), log(1+1), log(1+1)]
  
    du.reduce_weighted_logsumexp(x, w, axis=1)
    # ==> [log(-1+1+1), log(1+1+1)]
  
    du.reduce_weighted_logsumexp(x, w, axis=1, keep_dims=True)
    # ==> [[log(-1+1+1)], [log(1+1+1)]]
  
    du.reduce_weighted_logsumexp(x, w, axis=[0, 1])
    # ==> log(-1+5)
    ```
  
    Args:
      logx: The tensor to reduce. Should have numeric type.
      w: The weight tensor. Should have numeric type identical to `logx`.
      axis: The dimensions to reduce. If `None` (the default),
        reduces all dimensions. Must be in the range
        `[-rank(input_tensor), rank(input_tensor))`.
      keep_dims: If true, retains reduced dimensions with length 1.
      return_sign: If `True`, returns the sign of the result.
      name: A name for the operation (optional).
  
    Returns:
      lswe: The `log(abs(sum(weight * exp(x))))` reduced tensor.
      sign: (Optional) The sign of `sum(weight * exp(x))`.
    """
    with ops.name_scope(name, "reduce_weighted_logsumexp", [logx, w]):
      logx = ops.convert_to_tensor(logx, name="logx")
      if w is None:
        lswe = math_ops.reduce_logsumexp(logx, axis=axis, keepdims=keep_dims)
        if return_sign:
          sgn = array_ops.ones_like(lswe)
          return lswe, sgn
        return lswe
      w = ops.convert_to_tensor(w, dtype=logx.dtype, name="w")
      log_absw_x = logx + math_ops.log(math_ops.abs(w))
      max_log_absw_x = math_ops.reduce_max(log_absw_x, axis=axis, keepdims=True)
      # If the largest element is `-inf` or `inf` then we don't bother subtracting
      # off the max. We do this because otherwise we'd get `inf - inf = NaN`. That
      # this is ok follows from the fact that we're actually free to subtract any
      # value we like, so long as we add it back after taking the `log(sum(...))`.
      max_log_absw_x = array_ops.where(
          math_ops.is_inf(max_log_absw_x),
          array_ops.zeros_like(max_log_absw_x),
          max_log_absw_x)
      wx_over_max_absw_x = (
          math_ops.sign(w) * math_ops.exp(log_absw_x - max_log_absw_x))
      sum_wx_over_max_absw_x = math_ops.reduce_sum(
          wx_over_max_absw_x, axis=axis, keepdims=keep_dims)
      if not keep_dims:
        max_log_absw_x = array_ops.squeeze(max_log_absw_x, axis)
      sgn = math_ops.sign(sum_wx_over_max_absw_x)
      lswe = max_log_absw_x + math_ops.log(sgn * sum_wx_over_max_absw_x)
      if return_sign:
        return lswe, sgn
      return lswe
  
  
  # TODO(jvdillon): Merge this test back into:
  # tensorflow/python/ops/softplus_op_test.py
  # once TF core is accepting new ops.
  def softplus_inverse(x, name=None):
    """Computes the inverse softplus, i.e., x = softplus_inverse(softplus(x)).
  
    Mathematically this op is equivalent to:
  
    ```none
    softplus_inverse = log(exp(x) - 1.)
    ```
  
    Args:
      x: `Tensor`. Non-negative (not enforced), floating-point.
      name: A name for the operation (optional).
  
    Returns:
      `Tensor`. Has the same type/shape as input `x`.
    """
    with ops.name_scope(name, "softplus_inverse", values=[x]):
      x = ops.convert_to_tensor(x, name="x")
      # We begin by deriving a more numerically stable softplus_inverse:
      # x = softplus(y) = Log[1 + exp{y}], (which means x > 0).
      # ==> exp{x} = 1 + exp{y}                                (1)
      # ==> y = Log[exp{x} - 1]                                (2)
      #       = Log[(exp{x} - 1) / exp{x}] + Log[exp{x}]
      #       = Log[(1 - exp{-x}) / 1] + Log[exp{x}]
      #       = Log[1 - exp{-x}] + x                           (3)
      # (2) is the "obvious" inverse, but (3) is more stable than (2) for large x.
      # For small x (e.g. x = 1e-10), (3) will become -inf since 1 - exp{-x} will
      # be zero. To fix this, we use 1 - exp{-x} approx x for small x > 0.
      #
      # In addition to the numerically stable derivation above, we clamp
      # small/large values to be congruent with the logic in:
      # tensorflow/core/kernels/softplus_op.h
      #
      # Finally, we set the input to one whenever the input is too large or too
      # small. This ensures that no unchosen codepath is +/- inf. This is
      # necessary to ensure the gradient doesn't get NaNs. Recall that the
      # gradient of `where` behaves like `pred*pred_true + (1-pred)*pred_false`
      # thus an `inf` in an unselected path results in `0*inf=nan`. We are careful
      # to overwrite `x` with ones only when we will never actually use this
      # value. Note that we use ones and not zeros since `log(expm1(0.)) = -inf`.
      threshold = np.log(np.finfo(x.dtype.as_numpy_dtype).eps) + 2.
      is_too_small = math_ops.less(x, np.exp(threshold))
      is_too_large = math_ops.greater(x, -threshold)
      too_small_value = math_ops.log(x)
      too_large_value = x
      # This `where` will ultimately be a NOP because we won't select this
      # codepath whenever we used the surrogate `ones_like`.
      x = array_ops.where(math_ops.logical_or(is_too_small, is_too_large),
                          array_ops.ones_like(x), x)
      y = x + math_ops.log(-math_ops.expm1(-x))  # == log(expm1(x))
      return array_ops.where(is_too_small, too_small_value,
                             array_ops.where(is_too_large, too_large_value, y))
  
  
  # TODO(b/35290280): Add unit-tests.
  def dimension_size(x, axis):
    """Returns the size of a specific dimension."""
    # Since tf.gather isn't "constant-in, constant-out", we must first check the
    # static shape or fallback to dynamic shape.
    s = x.shape.with_rank_at_least(np.abs(axis))[axis].value
    if s is not None:
      return s
    return array_ops.shape(x)[axis]
  
  
  def process_quadrature_grid_and_probs(
      quadrature_grid_and_probs, dtype, validate_args, name=None):
    """Validates quadrature grid, probs or computes them as necessary.
  
    Args:
      quadrature_grid_and_probs: Python pair of `float`-like `Tensor`s
        representing the sample points and the corresponding (possibly
        normalized) weight.  When `None`, defaults to:
        `np.polynomial.hermite.hermgauss(deg=8)`.
      dtype: The expected `dtype` of `grid` and `probs`.
      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.
      name: Python `str` name prefixed to Ops created by this class.
  
    Returns:
       quadrature_grid_and_probs: Python pair of `float`-like `Tensor`s
        representing the sample points and the corresponding (possibly
        normalized) weight.
  
    Raises:
      ValueError: if `quadrature_grid_and_probs is not None` and
        `len(quadrature_grid_and_probs[0]) != len(quadrature_grid_and_probs[1])`
    """
    with ops.name_scope(name, "process_quadrature_grid_and_probs",
                        [quadrature_grid_and_probs]):
      if quadrature_grid_and_probs is None:
        grid, probs = np.polynomial.hermite.hermgauss(deg=8)
        grid = grid.astype(dtype.as_numpy_dtype)
        probs = probs.astype(dtype.as_numpy_dtype)
        probs /= np.linalg.norm(probs, ord=1, keepdims=True)
        grid = ops.convert_to_tensor(grid, name="grid", dtype=dtype)
        probs = ops.convert_to_tensor(probs, name="probs", dtype=dtype)
        return grid, probs
  
      grid, probs = tuple(quadrature_grid_and_probs)
      grid = ops.convert_to_tensor(grid, name="grid", dtype=dtype)
      probs = ops.convert_to_tensor(probs, name="unnormalized_probs",
                                    dtype=dtype)
      probs /= linalg_ops.norm(probs, ord=1, axis=-1, keepdims=True, name="probs")
  
      def _static_event_size(x):
        """Returns the static size of a specific dimension or `None`."""
        return x.shape.with_rank_at_least(1)[-1].value
  
      m, n = _static_event_size(probs), _static_event_size(grid)
      if m is not None and n is not None:
        if m != n:
          raise ValueError("`quadrature_grid_and_probs` must be a `tuple` of "
                           "same-length zero-th-dimension `Tensor`s "
                           "(saw lengths {}, {})".format(m, n))
      elif validate_args:
        assertions = [
            check_ops.assert_equal(
                dimension_size(probs, axis=-1),
                dimension_size(grid, axis=-1),
                message=("`quadrature_grid_and_probs` must be a `tuple` of "
                         "same-length zero-th-dimension `Tensor`s")),
        ]
        with ops.control_dependencies(assertions):
          grid = array_ops.identity(grid)
          probs = array_ops.identity(probs)
      return grid, probs
  
  
  def pad(x, axis, front=False, back=False, value=0, count=1, name=None):
    """Pads `value` to the front and/or back of a `Tensor` dim, `count` times.
  
    Args:
      x: `Tensor` input.
      axis: Scalar `int`-like `Tensor` representing the single dimension to pad.
        (Negative indexing is supported.)
      front: Python `bool`; if `True` the beginning of the `axis` dimension is
        padded with `value`, `count` times. If `False` no front padding is made.
      back: Python `bool`; if `True` the end of the `axis` dimension is
        padded with `value`, `count` times. If `False` no end padding is made.
      value: Scalar `int`-like `Tensor` representing the actual value added to the
        front and/or back of the `axis` dimension of `x`.
      count: Scalar `int`-like `Tensor` representing number of elements added to
        the front and/or back of the `axis` dimension of `x`. E.g., if
        `front = back = True` then `2 * count` elements are added.
      name: Python `str` name prefixed to Ops created by this function.
  
    Returns:
      pad: The padded version of input `x`.
  
    Raises:
      ValueError: if both `front` and `back` are `False`.
      TypeError: if `count` is not `int`-like.
    """
    with ops.name_scope(name, "pad", [x, value, count]):
      x = ops.convert_to_tensor(x, name="x")
      value = ops.convert_to_tensor(value, dtype=x.dtype, name="value")
      count = ops.convert_to_tensor(count, name="count")
      if not count.dtype.is_integer:
        raise TypeError("`count.dtype` (`{}`) must be `int`-like.".format(
            count.dtype.name))
      if not front and not back:
        raise ValueError("At least one of `front`, `back` must be `True`.")
      ndims = (x.shape.ndims if x.shape.ndims is not None
               else array_ops.rank(x, name="ndims"))
      axis = ops.convert_to_tensor(axis, name="axis")
      axis_ = tensor_util.constant_value(axis)
      if axis_ is not None:
        axis = axis_
        if axis < 0:
          axis = ndims + axis
        count_ = tensor_util.constant_value(count)
        if axis_ >= 0 or x.shape.ndims is not None:
          head = x.shape[:axis]
          middle = tensor_shape.TensorShape(
              None if count_ is None
              else (x.shape[axis] + count_ * (front + back)))
          tail = x.shape[axis+1:]
          final_shape = head.concatenate(middle.concatenate(tail))
        else:
          final_shape = None
      else:
        axis = array_ops.where(axis < 0, ndims + axis, axis)
        final_shape = None
      x = array_ops.pad(
          x,
          paddings=array_ops.one_hot(
              indices=array_ops.stack([axis if front else -1,
                                       axis if back else -1]),
              depth=ndims,
              axis=0,
              on_value=count,
              dtype=dtypes.int32),
          constant_values=value)
      if final_shape is not None:
        x.set_shape(final_shape)
      return x
  
  
  def parent_frame_arguments():
    """Returns parent frame arguments.
  
    When called inside a function, returns a dictionary with the caller's function
    arguments. These are positional arguments and keyword arguments (**kwargs),
    while variable arguments (*varargs) are excluded.
  
    When called at global scope, this will return an empty dictionary, since there
    are no arguments.
  
    WARNING: If caller function argument names are overloaded before invoking
    this method, then values will reflect the overloaded value. For this reason,
    we recommend calling `parent_frame_arguments` at the beginning of the
    function.
    """
    # All arguments and the names used for *varargs, and **kwargs
    arg_names, variable_arg_name, keyword_arg_name, local_vars = (
        tf_inspect._inspect.getargvalues(  # pylint: disable=protected-access
            # Get the first frame of the caller of this method.
            tf_inspect._inspect.stack()[1][0]))  # pylint: disable=protected-access
  
    # Remove the *varargs, and flatten the **kwargs. Both are
    # nested lists.
    local_vars.pop(variable_arg_name, {})
    keyword_args = local_vars.pop(keyword_arg_name, {})
  
    final_args = {}
    # Copy over arguments and their values. In general, local_vars
    # may contain more than just the arguments, since this method
    # can be called anywhere in a function.
    for arg_name in arg_names:
      final_args[arg_name] = local_vars.pop(arg_name)
    final_args.update(keyword_args)
  
    return final_args
  
  
  class AppendDocstring(object):
    """Helper class to promote private subclass docstring to public counterpart.
  
    Example:
  
    ```python
    class TransformedDistribution(Distribution):
      @distribution_util.AppendDocstring(
        additional_note="A special note!",
        kwargs_dict={"foo": "An extra arg."})
      def _prob(self, y, foo=None):
        pass
    ```
  
    In this case, the `AppendDocstring` decorator appends the `additional_note` to
    the docstring of `prob` (not `_prob`) and adds a new `kwargs`
    section with each dictionary item as a bullet-point.
  
    For a more detailed example, see `TransformedDistribution`.
    """
  
    def __init__(self, additional_note="", kwargs_dict=None):
      """Initializes the AppendDocstring object.
  
      Args:
        additional_note: Python string added as additional docstring to public
          version of function.
        kwargs_dict: Python string/string dictionary representing
          specific kwargs expanded from the **kwargs input.
  
      Raises:
        ValueError: if kwargs_dict.key contains whitespace.
        ValueError: if kwargs_dict.value contains newlines.
      """
      self._additional_note = additional_note
      if kwargs_dict:
        bullets = []
        for key in sorted(kwargs_dict.keys()):
          value = kwargs_dict[key]
          if any(x.isspace() for x in key):
            raise ValueError(
                "Parameter name \"%s\" contains whitespace." % key)
          value = value.lstrip()
          if "\n" in value:
            raise ValueError(
                "Parameter description for \"%s\" contains newlines." % key)
          bullets.append("*  `%s`: %s" % (key, value))
        self._additional_note += ("\n\n##### `kwargs`:\n\n" +
                                  "\n".join(bullets))
  
    def __call__(self, fn):
      @functools.wraps(fn)
      def _fn(*args, **kwargs):
        return fn(*args, **kwargs)
      if _fn.__doc__ is None:
        _fn.__doc__ = self._additional_note
      else:
        _fn.__doc__ += "\n%s" % self._additional_note
      return _fn
  