text: # Copyright 2017 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.
  # ==============================================================================
  """Vectorized Exponential distribution class, directly using LinearOperator."""
  
  from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  
  from tensorflow.contrib.distributions.python.ops import bijectors
  from tensorflow.contrib.distributions.python.ops import distribution_util
  from tensorflow.python.framework import ops
  from tensorflow.python.ops import array_ops
  from tensorflow.python.ops import math_ops
  from tensorflow.python.ops.distributions import exponential
  from tensorflow.python.ops.distributions import transformed_distribution
  from tensorflow.python.ops.linalg import linalg
  
  __all__ = ["VectorExponentialLinearOperator"]
  
  _mvn_sample_note = """
  `value` is a batch vector with compatible shape if `value` is a `Tensor` whose
  shape can be broadcast up to either:
  
  ```python
  self.batch_shape + self.event_shape
  ```
  
  or
  
  ```python
  [M1, ..., Mm] + self.batch_shape + self.event_shape
  ```
  
  """
  
  
  class VectorExponentialLinearOperator(
      transformed_distribution.TransformedDistribution):
    """The vectorization of the Exponential distribution on `R^k`.
  
    The vector exponential distribution is defined over a subset of `R^k`, and
    parameterized by a (batch of) length-`k` `loc` vector and a (batch of) `k x k`
    `scale` matrix:  `covariance = scale @ scale.T`, where `@` denotes
    matrix-multiplication.
  
    #### Mathematical Details
  
    The probability density function (pdf) is
  
    ```none
    pdf(y; loc, scale) = exp(-||x||_1) / Z,  for y in S(loc, scale),
    x = inv(scale) @ (y - loc),
    Z = |det(scale)|,
    ```
  
    where:
  
    * `loc` is a vector in `R^k`,
    * `scale` is a linear operator in `R^{k x k}`, `cov = scale @ scale.T`,
    * `S = {loc + scale @ x : x in R^k, x_1 > 0, ..., x_k > 0}`, is an image of
       the positive half-space,
    * `||x||_1` denotes the `l1` norm of `x`, `sum_i |x_i|`,
    * `Z` denotes the normalization constant.
  
    The VectorExponential distribution is a member of the [location-scale
    family](https://en.wikipedia.org/wiki/Location-scale_family), i.e., it can be
    constructed as,
  
    ```none
    X = (X_1, ..., X_k), each X_i ~ Exponential(rate=1)
    Y = (Y_1, ...,Y_k) = scale @ X + loc
    ```
  
    #### About `VectorExponential` and `Vector` distributions in TensorFlow.
  
    The `VectorExponential` is a non-standard distribution that has useful
    properties.
  
    The marginals `Y_1, ..., Y_k` are *not* Exponential random variables, due to
    the fact that the sum of Exponential random variables is not Exponential.
  
    Instead, `Y` is a vector whose components are linear combinations of
    Exponential random variables.  Thus, `Y` lives in the vector space generated
    by `vectors` of Exponential distributions.  This allows the user to decide the
    mean and covariance (by setting `loc` and `scale`), while preserving some
    properties of the Exponential distribution.  In particular, the tails of `Y_i`
    will be (up to polynomial factors) exponentially decaying.
  
    To see this last statement, note that the pdf of `Y_i` is the convolution of
    the pdf of `k` independent Exponential random variables.  One can then show by
    induction that distributions with exponential (up to polynomial factors) tails
    are closed under convolution.
  
  
    #### Examples
  
    ```python
    tfd = tf.contrib.distributions
  
    # Initialize a single 2-variate VectorExponential, supported on
    # {(x, y) in R^2 : x > 0, y > 0}.
    mat = [[1.0, 0.1],
           [0.1, 1.0]]
  
    vex = tfd.VectorExponentialLinearOperator(
        scale=tf.linalg.LinearOperatorFullMatrix(mat))
  
    # Compute the pdf of an`R^2` observation; return a scalar.
    vex.prob([1., 2.]).eval()  # shape: []
  
    # Initialize a 2-batch of 3-variate Vector Exponential's.
    mu = [[1., 2, 3],
          [1., 0, 0]]              # shape: [2, 3]
    scale_diag = [[1., 2, 3],
                  [0.5, 1, 1.5]]     # shape: [2, 3]
  
    vex = tfd.VectorExponentialLinearOperator(
        loc=mu,
        scale=tf.linalg.LinearOperatorDiag(scale_diag))
  
    # Compute the pdf of two `R^3` observations; return a length-2 vector.
    x = [[1.9, 2.2, 3.1],
         [10., 1.0, 9.0]]     # shape: [2, 3]
    vex.prob(x).eval()    # shape: [2]
    ```
  
    """
  
    def __init__(self,
                 loc=None,
                 scale=None,
                 validate_args=False,
                 allow_nan_stats=True,
                 name="VectorExponentialLinearOperator"):
      """Construct Vector Exponential distribution supported on a subset of `R^k`.
  
      The `batch_shape` is the broadcast shape between `loc` and `scale`
      arguments.
  
      The `event_shape` is given by last dimension of the matrix implied by
      `scale`. The last dimension of `loc` (if provided) must broadcast with this.
  
      Recall that `covariance = scale @ scale.T`.
  
      Additional leading dimensions (if any) will index batches.
  
      Args:
        loc: Floating-point `Tensor`. If this is set to `None`, `loc` is
          implicitly `0`. When specified, may have shape `[B1, ..., Bb, k]` where
          `b >= 0` and `k` is the event size.
        scale: Instance of `LinearOperator` with same `dtype` as `loc` and shape
          `[B1, ..., Bb, k, k]`.
        validate_args: Python `bool`, default `False`. Whether to validate input
          with asserts. If `validate_args` is `False`, and the inputs are
          invalid, correct behavior is not guaranteed.
        allow_nan_stats: Python `bool`, default `True`. If `False`, raise an
          exception if a statistic (e.g. mean/mode/etc...) is undefined for any
          batch member If `True`, batch members with valid parameters leading to
          undefined statistics will return NaN for this statistic.
        name: The name to give Ops created by the initializer.
  
      Raises:
        ValueError: if `scale` is unspecified.
        TypeError: if not `scale.dtype.is_floating`
      """
      parameters = distribution_util.parent_frame_arguments()
      if scale is None:
        raise ValueError("Missing required `scale` parameter.")
      if not scale.dtype.is_floating:
        raise TypeError("`scale` parameter must have floating-point dtype.")
  
      with ops.name_scope(name, values=[loc] + scale.graph_parents) as name:
        # Since expand_dims doesn't preserve constant-ness, we obtain the
        # non-dynamic value if possible.
        loc = ops.convert_to_tensor(loc, name="loc") if loc is not None else loc
        batch_shape, event_shape = distribution_util.shapes_from_loc_and_scale(
            loc, scale)
  
        super(VectorExponentialLinearOperator, self).__init__(
            distribution=exponential.Exponential(rate=array_ops.ones(
                [], dtype=scale.dtype), allow_nan_stats=allow_nan_stats),
            bijector=bijectors.AffineLinearOperator(
                shift=loc, scale=scale, validate_args=validate_args),
            batch_shape=batch_shape,
            event_shape=event_shape,
            validate_args=validate_args,
            name=name)
        self._parameters = parameters
  
    @property
    def loc(self):
      """The `loc` `Tensor` in `Y = scale @ X + loc`."""
      return self.bijector.shift
  
    @property
    def scale(self):
      """The `scale` `LinearOperator` in `Y = scale @ X + loc`."""
      return self.bijector.scale
  
    @distribution_util.AppendDocstring(_mvn_sample_note)
    def _log_prob(self, x):
      return super(VectorExponentialLinearOperator, self)._log_prob(x)
  
    @distribution_util.AppendDocstring(_mvn_sample_note)
    def _prob(self, x):
      return super(VectorExponentialLinearOperator, self)._prob(x)
  
    def _mean(self):
      # Let
      #   W = (w1,...,wk), with wj ~ iid Exponential(0, 1).
      # Then this distribution is
      #   X = loc + LW,
      # and then E[X] = loc + L1, where 1 is the vector of ones.
      scale_x_ones = self.bijector.scale.matvec(
          array_ops.ones(self._mode_mean_shape(), self.dtype))
  
      if self.loc is None:
        return scale_x_ones
  
      return array_ops.identity(self.loc) + scale_x_ones
  
    def _covariance(self):
      # Let
      #   W = (w1,...,wk), with wj ~ iid Exponential(0, 1).
      # Then this distribution is
      #   X = loc + LW,
      # and then since Cov(wi, wj) = 1 if i=j, and 0 otherwise,
      #   Cov(X) = L Cov(W W^T) L^T = L L^T.
      if distribution_util.is_diagonal_scale(self.scale):
        return array_ops.matrix_diag(math_ops.square(self.scale.diag_part()))
      else:
        return self.scale.matmul(self.scale.to_dense(), adjoint_arg=True)
  
    def _variance(self):
      if distribution_util.is_diagonal_scale(self.scale):
        return math_ops.square(self.scale.diag_part())
      elif (isinstance(self.scale, linalg.LinearOperatorLowRankUpdate) and
            self.scale.is_self_adjoint):
        return array_ops.matrix_diag_part(
            self.scale.matmul(self.scale.to_dense()))
      else:
        return array_ops.matrix_diag_part(
            self.scale.matmul(self.scale.to_dense(), adjoint_arg=True))
  
    def _stddev(self):
      if distribution_util.is_diagonal_scale(self.scale):
        return math_ops.abs(self.scale.diag_part())
      elif (isinstance(self.scale, linalg.LinearOperatorLowRankUpdate) and
            self.scale.is_self_adjoint):
        return math_ops.sqrt(
            array_ops.matrix_diag_part(self.scale.matmul(self.scale.to_dense())))
      else:
        return math_ops.sqrt(
            array_ops.matrix_diag_part(
                self.scale.matmul(self.scale.to_dense(), adjoint_arg=True)))
  
    def _mode(self):
      scale_x_zeros = self.bijector.scale.matvec(
          array_ops.zeros(self._mode_mean_shape(), self.dtype))
  
      if self.loc is None:
        return scale_x_zeros
  
      return array_ops.identity(self.loc) + scale_x_zeros
  
    def _mode_mean_shape(self):
      """Shape for the mode/mean Tensors."""
      shape = self.batch_shape.concatenate(self.event_shape)
      has_static_shape = shape.is_fully_defined()
      if not has_static_shape:
        shape = array_ops.concat([
            self.batch_shape_tensor(),
            self.event_shape_tensor(),
        ], 0)
      return shape
  