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.
  # ==============================================================================
  """Multivariate Normal distribution class initialized with a full covariance."""
  
  from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  
  from tensorflow.contrib.distributions.python.ops import mvn_tril
  from tensorflow.python.framework import ops
  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
  
  
  __all__ = [
      "MultivariateNormalFullCovariance",
  ]
  
  
  class MultivariateNormalFullCovariance(mvn_tril.MultivariateNormalTriL):
    """The multivariate normal distribution on `R^k`.
  
    The Multivariate Normal distribution is defined over `R^k` and parameterized
    by a (batch of) length-`k` `loc` vector (aka "mu") and a (batch of) `k x k`
    `covariance_matrix` matrices that are the covariance.
    This is different than the other multivariate normals, which are parameterized
    by a matrix more akin to the standard deviation.
  
    #### Mathematical Details
  
    The probability density function (pdf) is, with `@` as matrix multiplication,
  
    ```none
    pdf(x; loc, covariance_matrix) = exp(-0.5 y) / Z,
    y = (x - loc)^T @ inv(covariance_matrix) @ (x - loc)
    Z = (2 pi)**(0.5 k) |det(covariance_matrix)|**(0.5).
    ```
  
    where:
  
    * `loc` is a vector in `R^k`,
    * `covariance_matrix` is an `R^{k x k}` symmetric positive definite matrix,
    * `Z` denotes the normalization constant.
  
    Additional leading dimensions (if any) in `loc` and `covariance_matrix` allow
    for batch dimensions.
  
    The MultivariateNormal distribution is a member of the [location-scale
    family](https://en.wikipedia.org/wiki/Location-scale_family), i.e., it can be
    constructed e.g. as,
  
    ```none
    X ~ MultivariateNormal(loc=0, scale=1)   # Identity scale, zero shift.
    scale = Cholesky(covariance_matrix)
    Y = scale @ X + loc
    ```
  
    #### Examples
  
    ```python
    tfd = tf.contrib.distributions
  
    # Initialize a single 3-variate Gaussian.
    mu = [1., 2, 3]
    cov = [[ 0.36,  0.12,  0.06],
           [ 0.12,  0.29, -0.13],
           [ 0.06, -0.13,  0.26]]
    mvn = tfd.MultivariateNormalFullCovariance(
        loc=mu,
        covariance_matrix=cov)
  
    mvn.mean().eval()
    # ==> [1., 2, 3]
  
    # Covariance agrees with covariance_matrix.
    mvn.covariance().eval()
    # ==> [[ 0.36,  0.12,  0.06],
    #      [ 0.12,  0.29, -0.13],
    #      [ 0.06, -0.13,  0.26]]
  
    # Compute the pdf of an observation in `R^3` ; return a scalar.
    mvn.prob([-1., 0, 1]).eval()  # shape: []
  
    # Initialize a 2-batch of 3-variate Gaussians.
    mu = [[1., 2, 3],
          [11, 22, 33]]              # shape: [2, 3]
    covariance_matrix = ...  # shape: [2, 3, 3], symmetric, positive definite.
    mvn = tfd.MultivariateNormalFullCovariance(
        loc=mu,
        covariance=covariance_matrix)
  
    # Compute the pdf of two `R^3` observations; return a length-2 vector.
    x = [[-0.9, 0, 0.1],
         [-10, 0, 9]]     # shape: [2, 3]
    mvn.prob(x).eval()    # shape: [2]
  
    ```
  
    """
  
    def __init__(self,
                 loc=None,
                 covariance_matrix=None,
                 validate_args=False,
                 allow_nan_stats=True,
                 name="MultivariateNormalFullCovariance"):
      """Construct Multivariate Normal distribution on `R^k`.
  
      The `batch_shape` is the broadcast shape between `loc` and
      `covariance_matrix` arguments.
  
      The `event_shape` is given by last dimension of the matrix implied by
      `covariance_matrix`. The last dimension of `loc` (if provided) must
      broadcast with this.
  
      A non-batch `covariance_matrix` matrix is a `k x k` symmetric positive
      definite matrix.  In other words it is (real) symmetric with all eigenvalues
      strictly positive.
  
      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.
        covariance_matrix: Floating-point, symmetric positive definite `Tensor` of
          same `dtype` as `loc`.  The strict upper triangle of `covariance_matrix`
          is ignored, so if `covariance_matrix` is not symmetric no error will be
          raised (unless `validate_args is True`).  `covariance_matrix` has shape
          `[B1, ..., Bb, k, k]` where `b >= 0` and `k` is the event size.
        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.
        name: Python `str` name prefixed to Ops created by this class.
  
      Raises:
        ValueError: if neither `loc` nor `covariance_matrix` are specified.
      """
      parameters = dict(locals())
  
      # Convert the covariance_matrix up to a scale_tril and call MVNTriL.
      with ops.name_scope(name) as name:
        with ops.name_scope("init", values=[loc, covariance_matrix]):
          if covariance_matrix is None:
            scale_tril = None
          else:
            covariance_matrix = ops.convert_to_tensor(
                covariance_matrix, name="covariance_matrix")
            if validate_args:
              covariance_matrix = control_flow_ops.with_dependencies([
                  check_ops.assert_near(
                      covariance_matrix,
                      array_ops.matrix_transpose(covariance_matrix),
                      message="Matrix was not symmetric")], covariance_matrix)
            # No need to validate that covariance_matrix is non-singular.
            # LinearOperatorLowerTriangular has an assert_non_singular method that
            # is called by the Bijector.
            # However, cholesky() ignores the upper triangular part, so we do need
            # to separately assert symmetric.
            scale_tril = linalg_ops.cholesky(covariance_matrix)
          super(MultivariateNormalFullCovariance, self).__init__(
              loc=loc,
              scale_tril=scale_tril,
              validate_args=validate_args,
              allow_nan_stats=allow_nan_stats,
              name=name)
      self._parameters = parameters
  