vector_laplace_diag.py

# Copyright 2017 The TensorFlow Authors. All Rights Reserved.
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# ==============================================================================
"""Distribution of a vectorized Laplace, with uncorrelated components."""

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

from tensorflow.contrib.distributions.python.ops import distribution_util
from tensorflow.contrib.distributions.python.ops import vector_laplace_linear_operator as vector_laplace_linop
from tensorflow.python.framework import ops
from tensorflow.python.util import deprecation


__all__ = [
    "VectorLaplaceDiag",
]


class VectorLaplaceDiag(
    vector_laplace_linop.VectorLaplaceLinearOperator):
  """The vectorization of the Laplace distribution on `R^k`.

  The vector laplace distribution is defined over `R^k`, and parameterized by
  a (batch of) length-`k` `loc` vector (the means) and a (batch of) `k x k`
  `scale` matrix:  `covariance = 2 * scale @ scale.T`, where `@` denotes
  matrix-multiplication.

  #### Mathematical Details

  The probability density function (pdf) is,

  ```none
  pdf(x; loc, scale) = exp(-||y||_1) / Z,
  y = inv(scale) @ (x - loc),
  Z = 2**k |det(scale)|,
  ```

  where:

  * `loc` is a vector in `R^k`,
  * `scale` is a linear operator in `R^{k x k}`, `cov = scale @ scale.T`,
  * `Z` denotes the normalization constant, and,
  * `||y||_1` denotes the `l1` norm of `y`, `sum_i |y_i|.

  A (non-batch) `scale` matrix is:

  ```none
  scale = diag(scale_diag + scale_identity_multiplier * ones(k))
  ```

  where:

  * `scale_diag.shape = [k]`, and,
  * `scale_identity_multiplier.shape = []`.

  Additional leading dimensions (if any) will index batches.

  If both `scale_diag` and `scale_identity_multiplier` are `None`, then
  `scale` is the Identity matrix.

  The VectorLaplace 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 ~ Laplace(loc=0, scale=1)
  Y = (Y_1, ...,Y_k) = scale @ X + loc
  ```

  #### About `VectorLaplace` and `Vector` distributions in TensorFlow.

  The `VectorLaplace` is a non-standard distribution that has useful properties.

  The marginals `Y_1, ..., Y_k` are *not* Laplace random variables, due to
  the fact that the sum of Laplace random variables is not Laplace.

  Instead, `Y` is a vector whose components are linear combinations of Laplace
  random variables.  Thus, `Y` lives in the vector space generated by `vectors`
  of Laplace distributions.  This allows the user to decide the mean and
  covariance (by setting `loc` and `scale`), while preserving some properties of
  the Laplace 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 Laplace 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 VectorLaplace.
  vla = tfd.VectorLaplaceDiag(
      loc=[1., -1],
      scale_diag=[1, 2.])

  vla.mean().eval()
  # ==> [1., -1]

  vla.stddev().eval()
  # ==> [1., 2] * sqrt(2)

  # Evaluate this on an observation in `R^2`, returning a scalar.
  vla.prob([-1., 0]).eval()  # shape: []

  # Initialize a 3-batch, 2-variate scaled-identity VectorLaplace.
  vla = tfd.VectorLaplaceDiag(
      loc=[1., -1],
      scale_identity_multiplier=[1, 2., 3])

  vla.mean().eval()  # shape: [3, 2]
  # ==> [[1., -1]
  #      [1, -1],
  #      [1, -1]]

  vla.stddev().eval()  # shape: [3, 2]
  # ==> sqrt(2) * [[1., 1],
  #                [2, 2],
  #                [3, 3]]

  # Evaluate this on an observation in `R^2`, returning a length-3 vector.
  vla.prob([-1., 0]).eval()  # shape: [3]

  # Initialize a 2-batch of 3-variate VectorLaplace's.
  vla = tfd.VectorLaplaceDiag(
      loc=[[1., 2, 3],
           [11, 22, 33]]           # shape: [2, 3]
      scale_diag=[[1., 2, 3],
                  [0.5, 1, 1.5]])  # shape: [2, 3]

  # Evaluate this on a two observations, each in `R^3`, returning a length-2
  # vector.
  x = [[-1., 0, 1],
       [-11, 0, 11.]]   # shape: [2, 3].
  vla.prob(x).eval()    # shape: [2]
  ```

  """

  @deprecation.deprecated(
      "2018-10-01",
      "The TensorFlow Distributions library has moved to "
      "TensorFlow Probability "
      "(https://github.com/tensorflow/probability). You "
      "should update all references to use `tfp.distributions` "
      "instead of `tf.contrib.distributions`.",
      warn_once=True)
  def __init__(self,
               loc=None,
               scale_diag=None,
               scale_identity_multiplier=None,
               validate_args=False,
               allow_nan_stats=True,
               name="VectorLaplaceDiag"):
    """Construct Vector Laplace distribution on `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 = 2 * scale @ scale.T`.

    ```none
    scale = diag(scale_diag + scale_identity_multiplier * ones(k))
    ```

    where:

    * `scale_diag.shape = [k]`, and,
    * `scale_identity_multiplier.shape = []`.

    Additional leading dimensions (if any) will index batches.

    If both `scale_diag` and `scale_identity_multiplier` are `None`, then
    `scale` is the Identity matrix.

    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_diag: Non-zero, floating-point `Tensor` representing a diagonal
        matrix added to `scale`. May have shape `[B1, ..., Bb, k]`, `b >= 0`,
        and characterizes `b`-batches of `k x k` diagonal matrices added to
        `scale`. When both `scale_identity_multiplier` and `scale_diag` are
        `None` then `scale` is the `Identity`.
      scale_identity_multiplier: Non-zero, floating-point `Tensor` representing
        a scaled-identity-matrix added to `scale`. May have shape
        `[B1, ..., Bb]`, `b >= 0`, and characterizes `b`-batches of scaled
        `k x k` identity matrices added to `scale`. When both
        `scale_identity_multiplier` and `scale_diag` are `None` then `scale` is
        the `Identity`.
      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 at most `scale_identity_multiplier` is specified.
    """
    parameters = dict(locals())
    with ops.name_scope(name):
      with ops.name_scope("init", values=[
          loc, scale_diag, scale_identity_multiplier]):
        # No need to validate_args while making diag_scale.  The returned
        # LinearOperatorDiag has an assert_non_singular method that is called by
        # the Bijector.
        scale = distribution_util.make_diag_scale(
            loc=loc,
            scale_diag=scale_diag,
            scale_identity_multiplier=scale_identity_multiplier,
            validate_args=False,
            assert_positive=False)
    super(VectorLaplaceDiag, self).__init__(
        loc=loc,
        scale=scale,
        validate_args=validate_args,
        allow_nan_stats=allow_nan_stats,
        name=name)
    self._parameters = parameters