dirichlet.py

# 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
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# ==============================================================================
"""The Dirichlet distribution class."""

import numpy as np

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 math_ops
from tensorflow.python.ops import random_ops
from tensorflow.python.ops import special_math_ops
from tensorflow.python.ops.distributions import distribution
from tensorflow.python.ops.distributions import kullback_leibler
from tensorflow.python.ops.distributions import util as distribution_util
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export


__all__ = [
    "Dirichlet",
]


_dirichlet_sample_note = """Note: `value` must be a non-negative tensor with
dtype `self.dtype` and be in the `(self.event_shape() - 1)`-simplex, i.e.,
`tf.reduce_sum(value, -1) = 1`. It must have a shape compatible with
`self.batch_shape() + self.event_shape()`."""


@tf_export(v1=["distributions.Dirichlet"])
class Dirichlet(distribution.Distribution):
  """Dirichlet distribution.

  The Dirichlet distribution is defined over the
  [`(k-1)`-simplex](https://en.wikipedia.org/wiki/Simplex) using a positive,
  length-`k` vector `concentration` (`k > 1`). The Dirichlet is identically the
  Beta distribution when `k = 2`.

  #### Mathematical Details

  The Dirichlet is a distribution over the open `(k-1)`-simplex, i.e.,

  ```none
  S^{k-1} = { (x_0, ..., x_{k-1}) in R^k : sum_j x_j = 1 and all_j x_j > 0 }.
  ```

  The probability density function (pdf) is,

  ```none
  pdf(x; alpha) = prod_j x_j**(alpha_j - 1) / Z
  Z = prod_j Gamma(alpha_j) / Gamma(sum_j alpha_j)
  ```

  where:

  * `x in S^{k-1}`, i.e., the `(k-1)`-simplex,
  * `concentration = alpha = [alpha_0, ..., alpha_{k-1}]`, `alpha_j > 0`,
  * `Z` is the normalization constant aka the [multivariate beta function](
    https://en.wikipedia.org/wiki/Beta_function#Multivariate_beta_function),
    and,
  * `Gamma` is the [gamma function](
    https://en.wikipedia.org/wiki/Gamma_function).

  The `concentration` represents mean total counts of class occurrence, i.e.,

  ```none
  concentration = alpha = mean * total_concentration
  ```

  where `mean` in `S^{k-1}` and `total_concentration` is a positive real number
  representing a mean total count.

  Distribution parameters are automatically broadcast in all functions; see
  examples for details.

  Warning: Some components of the samples can be zero due to finite precision.
  This happens more often when some of the concentrations are very small.
  Make sure to round the samples to `np.finfo(dtype).tiny` before computing the
  density.

  Samples of this distribution are reparameterized (pathwise differentiable).
  The derivatives are computed using the approach described in
  (Figurnov et al., 2018).

  #### Examples

  ```python
  import tensorflow_probability as tfp
  tfd = tfp.distributions

  # Create a single trivariate Dirichlet, with the 3rd class being three times
  # more frequent than the first. I.e., batch_shape=[], event_shape=[3].
  alpha = [1., 2, 3]
  dist = tfd.Dirichlet(alpha)

  dist.sample([4, 5])  # shape: [4, 5, 3]

  # x has one sample, one batch, three classes:
  x = [.2, .3, .5]   # shape: [3]
  dist.prob(x)       # shape: []

  # x has two samples from one batch:
  x = [[.1, .4, .5],
       [.2, .3, .5]]
  dist.prob(x)         # shape: [2]

  # alpha will be broadcast to shape [5, 7, 3] to match x.
  x = [[...]]   # shape: [5, 7, 3]
  dist.prob(x)  # shape: [5, 7]
  ```

  ```python
  # Create batch_shape=[2], event_shape=[3]:
  alpha = [[1., 2, 3],
           [4, 5, 6]]   # shape: [2, 3]
  dist = tfd.Dirichlet(alpha)

  dist.sample([4, 5])  # shape: [4, 5, 2, 3]

  x = [.2, .3, .5]
  # x will be broadcast as [[.2, .3, .5],
  #                         [.2, .3, .5]],
  # thus matching batch_shape [2, 3].
  dist.prob(x)         # shape: [2]
  ```

  Compute the gradients of samples w.r.t. the parameters:

  ```python
  alpha = tf.constant([1.0, 2.0, 3.0])
  dist = tfd.Dirichlet(alpha)
  samples = dist.sample(5)  # Shape [5, 3]
  loss = tf.reduce_mean(tf.square(samples))  # Arbitrary loss function
  # Unbiased stochastic gradients of the loss function
  grads = tf.gradients(loss, alpha)
  ```

  References:
    Implicit Reparameterization Gradients:
      [Figurnov et al., 2018]
      (http://papers.nips.cc/paper/7326-implicit-reparameterization-gradients)
      ([pdf]
      (http://papers.nips.cc/paper/7326-implicit-reparameterization-gradients.pdf))
  """

  @deprecation.deprecated(
      "2019-01-01",
      "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.distributions`.",
      warn_once=True)
  def __init__(self,
               concentration,
               validate_args=False,
               allow_nan_stats=True,
               name="Dirichlet"):
    """Initialize a batch of Dirichlet distributions.

    Args:
      concentration: Positive floating-point `Tensor` indicating mean number
        of class occurrences; aka "alpha". Implies `self.dtype`, and
        `self.batch_shape`, `self.event_shape`, i.e., if
        `concentration.shape = [N1, N2, ..., Nm, k]` then
        `batch_shape = [N1, N2, ..., Nm]` and
        `event_shape = [k]`.
      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.
    """
    parameters = dict(locals())
    with ops.name_scope(name, values=[concentration]) as name:
      self._concentration = self._maybe_assert_valid_concentration(
          ops.convert_to_tensor(concentration, name="concentration"),
          validate_args)
      self._total_concentration = math_ops.reduce_sum(self._concentration, -1)
    super(Dirichlet, self).__init__(
        dtype=self._concentration.dtype,
        validate_args=validate_args,
        allow_nan_stats=allow_nan_stats,
        reparameterization_type=distribution.FULLY_REPARAMETERIZED,
        parameters=parameters,
        graph_parents=[self._concentration,
                       self._total_concentration],
        name=name)

  @property
  def concentration(self):
    """Concentration parameter; expected counts for that coordinate."""
    return self._concentration

  @property
  def total_concentration(self):
    """Sum of last dim of concentration parameter."""
    return self._total_concentration

  def _batch_shape_tensor(self):
    return array_ops.shape(self.total_concentration)

  def _batch_shape(self):
    return self.total_concentration.get_shape()

  def _event_shape_tensor(self):
    return array_ops.shape(self.concentration)[-1:]

  def _event_shape(self):
    return self.concentration.get_shape().with_rank_at_least(1)[-1:]

  def _sample_n(self, n, seed=None):
    gamma_sample = random_ops.random_gamma(
        shape=[n],
        alpha=self.concentration,
        dtype=self.dtype,
        seed=seed)
    return gamma_sample / math_ops.reduce_sum(gamma_sample, -1, keepdims=True)

  @distribution_util.AppendDocstring(_dirichlet_sample_note)
  def _log_prob(self, x):
    return self._log_unnormalized_prob(x) - self._log_normalization()

  @distribution_util.AppendDocstring(_dirichlet_sample_note)
  def _prob(self, x):
    return math_ops.exp(self._log_prob(x))

  def _log_unnormalized_prob(self, x):
    x = self._maybe_assert_valid_sample(x)
    return math_ops.reduce_sum(math_ops.xlogy(self.concentration - 1., x), -1)

  def _log_normalization(self):
    return special_math_ops.lbeta(self.concentration)

  def _entropy(self):
    k = math_ops.cast(self.event_shape_tensor()[0], self.dtype)
    return (
        self._log_normalization()
        + ((self.total_concentration - k)
           * math_ops.digamma(self.total_concentration))
        - math_ops.reduce_sum(
            (self.concentration - 1.) * math_ops.digamma(self.concentration),
            axis=-1))

  def _mean(self):
    return self.concentration / self.total_concentration[..., array_ops.newaxis]

  def _covariance(self):
    x = self._variance_scale_term() * self._mean()
    # pylint: disable=invalid-unary-operand-type
    return array_ops.matrix_set_diag(
        -math_ops.matmul(
            x[..., array_ops.newaxis],
            x[..., array_ops.newaxis, :]),  # outer prod
        self._variance())

  def _variance(self):
    scale = self._variance_scale_term()
    x = scale * self._mean()
    return x * (scale - x)

  def _variance_scale_term(self):
    """Helper to `_covariance` and `_variance` which computes a shared scale."""
    return math_ops.rsqrt(1. + self.total_concentration[..., array_ops.newaxis])

  @distribution_util.AppendDocstring(
      """Note: The mode is undefined when any `concentration <= 1`. If
      `self.allow_nan_stats` is `True`, `NaN` is used for undefined modes. If
      `self.allow_nan_stats` is `False` an exception is raised when one or more
      modes are undefined.""")
  def _mode(self):
    k = math_ops.cast(self.event_shape_tensor()[0], self.dtype)
    mode = (self.concentration - 1.) / (
        self.total_concentration[..., array_ops.newaxis] - k)
    if self.allow_nan_stats:
      nan = array_ops.fill(
          array_ops.shape(mode),
          np.array(np.nan, dtype=self.dtype.as_numpy_dtype()),
          name="nan")
      return array_ops.where_v2(
          math_ops.reduce_all(self.concentration > 1., axis=-1), mode, nan)
    return control_flow_ops.with_dependencies([
        check_ops.assert_less(
            array_ops.ones([], self.dtype),
            self.concentration,
            message="Mode undefined when any concentration <= 1"),
    ], mode)

  def _maybe_assert_valid_concentration(self, concentration, validate_args):
    """Checks the validity of the concentration parameter."""
    if not validate_args:
      return concentration
    return control_flow_ops.with_dependencies([
        check_ops.assert_positive(
            concentration,
            message="Concentration parameter must be positive."),
        check_ops.assert_rank_at_least(
            concentration, 1,
            message="Concentration parameter must have >=1 dimensions."),
        check_ops.assert_less(
            1, array_ops.shape(concentration)[-1],
            message="Concentration parameter must have event_size >= 2."),
    ], concentration)

  def _maybe_assert_valid_sample(self, x):
    """Checks the validity of a sample."""
    if not self.validate_args:
      return x
    return control_flow_ops.with_dependencies([
        check_ops.assert_positive(x, message="samples must be positive"),
        check_ops.assert_near(
            array_ops.ones([], dtype=self.dtype),
            math_ops.reduce_sum(x, -1),
            message="sample last-dimension must sum to `1`"),
    ], x)


@kullback_leibler.RegisterKL(Dirichlet, Dirichlet)
def _kl_dirichlet_dirichlet(d1, d2, name=None):
  """Batchwise KL divergence KL(d1 || d2) with d1 and d2 Dirichlet.

  Args:
    d1: instance of a Dirichlet distribution object.
    d2: instance of a Dirichlet distribution object.
    name: (optional) Name to use for created operations.
      default is "kl_dirichlet_dirichlet".

  Returns:
    Batchwise KL(d1 || d2)
  """
  with ops.name_scope(name, "kl_dirichlet_dirichlet", values=[
      d1.concentration, d2.concentration]):
    # The KL between Dirichlet distributions can be derived as follows. We have
    #
    #   Dir(x; a) = 1 / B(a) * prod_i[x[i]^(a[i] - 1)]
    #
    # where B(a) is the multivariate Beta function:
    #
    #   B(a) = Gamma(a[1]) * ... * Gamma(a[n]) / Gamma(a[1] + ... + a[n])
    #
    # The KL is
    #
    #   KL(Dir(x; a), Dir(x; b)) = E_Dir(x; a){log(Dir(x; a) / Dir(x; b))}
    #
    # so we'll need to know the log density of the Dirichlet. This is
    #
    #   log(Dir(x; a)) = sum_i[(a[i] - 1) log(x[i])] - log B(a)
    #
    # The only term that matters for the expectations is the log(x[i]). To
    # compute the expectation of this term over the Dirichlet density, we can
    # use the following facts about the Dirichlet in exponential family form:
    #   1. log(x[i]) is a sufficient statistic
    #   2. expected sufficient statistics (of any exp family distribution) are
    #      equal to derivatives of the log normalizer with respect to
    #      corresponding natural parameters: E{T[i](x)} = dA/d(eta[i])
    #
    # To proceed, we can rewrite the Dirichlet density in exponential family
    # form as follows:
    #
    #   Dir(x; a) = exp{eta(a) . T(x) - A(a)}
    #
    # where '.' is the dot product of vectors eta and T, and A is a scalar:
    #
    #   eta[i](a) = a[i] - 1
    #     T[i](x) = log(x[i])
    #        A(a) = log B(a)
    #
    # Now, we can use fact (2) above to write
    #
    #   E_Dir(x; a)[log(x[i])]
    #       = dA(a) / da[i]
    #       = d/da[i] log B(a)
    #       = d/da[i] (sum_j lgamma(a[j])) - lgamma(sum_j a[j])
    #       = digamma(a[i])) - digamma(sum_j a[j])
    #
    # Putting it all together, we have
    #
    # KL[Dir(x; a) || Dir(x; b)]
    #     = E_Dir(x; a){log(Dir(x; a) / Dir(x; b)}
    #     = E_Dir(x; a){sum_i[(a[i] - b[i]) log(x[i])} - (lbeta(a) - lbeta(b))
    #     = sum_i[(a[i] - b[i]) * E_Dir(x; a){log(x[i])}] - lbeta(a) + lbeta(b)
    #     = sum_i[(a[i] - b[i]) * (digamma(a[i]) - digamma(sum_j a[j]))]
    #          - lbeta(a) + lbeta(b))

    digamma_sum_d1 = math_ops.digamma(
        math_ops.reduce_sum(d1.concentration, axis=-1, keepdims=True))
    digamma_diff = math_ops.digamma(d1.concentration) - digamma_sum_d1
    concentration_diff = d1.concentration - d2.concentration

    return (math_ops.reduce_sum(concentration_diff * digamma_diff, axis=-1) -
            special_math_ops.lbeta(d1.concentration) +
            special_math_ops.lbeta(d2.concentration))