323 lines
11 KiB
Python
323 lines
11 KiB
Python
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# Copyright 2016 The TensorFlow Authors. All Rights Reserved.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# http://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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# ==============================================================================
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"""The Gamma distribution class."""
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from __future__ import absolute_import
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from __future__ import division
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from __future__ import print_function
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import numpy as np
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from tensorflow.python.framework import constant_op
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from tensorflow.python.framework import dtypes
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from tensorflow.python.framework import ops
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from tensorflow.python.framework import tensor_shape
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from tensorflow.python.ops import array_ops
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from tensorflow.python.ops import check_ops
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from tensorflow.python.ops import control_flow_ops
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from tensorflow.python.ops import math_ops
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from tensorflow.python.ops import nn
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from tensorflow.python.ops import random_ops
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from tensorflow.python.ops.distributions import distribution
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from tensorflow.python.ops.distributions import kullback_leibler
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from tensorflow.python.ops.distributions import util as distribution_util
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from tensorflow.python.util.tf_export import tf_export
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__all__ = [
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"Gamma",
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"GammaWithSoftplusConcentrationRate",
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]
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@tf_export("distributions.Gamma")
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class Gamma(distribution.Distribution):
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"""Gamma distribution.
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The Gamma distribution is defined over positive real numbers using
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parameters `concentration` (aka "alpha") and `rate` (aka "beta").
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#### Mathematical Details
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The probability density function (pdf) is,
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```none
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pdf(x; alpha, beta, x > 0) = x**(alpha - 1) exp(-x beta) / Z
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Z = Gamma(alpha) beta**(-alpha)
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```
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where:
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* `concentration = alpha`, `alpha > 0`,
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* `rate = beta`, `beta > 0`,
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* `Z` is the normalizing constant, and,
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* `Gamma` is the [gamma function](
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https://en.wikipedia.org/wiki/Gamma_function).
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The cumulative density function (cdf) is,
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```none
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cdf(x; alpha, beta, x > 0) = GammaInc(alpha, beta x) / Gamma(alpha)
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```
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where `GammaInc` is the [lower incomplete Gamma function](
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https://en.wikipedia.org/wiki/Incomplete_gamma_function).
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The parameters can be intuited via their relationship to mean and stddev,
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```none
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concentration = alpha = (mean / stddev)**2
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rate = beta = mean / stddev**2 = concentration / mean
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```
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Distribution parameters are automatically broadcast in all functions; see
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examples for details.
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Warning: The samples of this distribution are always non-negative. However,
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the samples that are smaller than `np.finfo(dtype).tiny` are rounded
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to this value, so it appears more often than it should.
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This should only be noticeable when the `concentration` is very small, or the
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`rate` is very large. See note in `tf.random_gamma` docstring.
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Samples of this distribution are reparameterized (pathwise differentiable).
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The derivatives are computed using the approach described in the paper
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[Michael Figurnov, Shakir Mohamed, Andriy Mnih.
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Implicit Reparameterization Gradients, 2018](https://arxiv.org/abs/1805.08498)
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#### Examples
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```python
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dist = tf.distributions.Gamma(concentration=3.0, rate=2.0)
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dist2 = tf.distributions.Gamma(concentration=[3.0, 4.0], rate=[2.0, 3.0])
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```
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Compute the gradients of samples w.r.t. the parameters:
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```python
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concentration = tf.constant(3.0)
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rate = tf.constant(2.0)
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dist = tf.distributions.Gamma(concentration, rate)
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samples = dist.sample(5) # Shape [5]
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loss = tf.reduce_mean(tf.square(samples)) # Arbitrary loss function
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# Unbiased stochastic gradients of the loss function
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grads = tf.gradients(loss, [concentration, rate])
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```
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"""
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def __init__(self,
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concentration,
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rate,
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validate_args=False,
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allow_nan_stats=True,
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name="Gamma"):
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"""Construct Gamma with `concentration` and `rate` parameters.
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The parameters `concentration` and `rate` must be shaped in a way that
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supports broadcasting (e.g. `concentration + rate` is a valid operation).
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Args:
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concentration: Floating point tensor, the concentration params of the
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distribution(s). Must contain only positive values.
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rate: Floating point tensor, the inverse scale params of the
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distribution(s). Must contain only positive values.
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validate_args: Python `bool`, default `False`. When `True` distribution
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parameters are checked for validity despite possibly degrading runtime
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performance. When `False` invalid inputs may silently render incorrect
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outputs.
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allow_nan_stats: Python `bool`, default `True`. When `True`, statistics
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(e.g., mean, mode, variance) use the value "`NaN`" to indicate the
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result is undefined. When `False`, an exception is raised if one or
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more of the statistic's batch members are undefined.
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name: Python `str` name prefixed to Ops created by this class.
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Raises:
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TypeError: if `concentration` and `rate` are different dtypes.
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"""
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parameters = dict(locals())
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with ops.name_scope(name, values=[concentration, rate]) as name:
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with ops.control_dependencies([
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check_ops.assert_positive(concentration),
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check_ops.assert_positive(rate),
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] if validate_args else []):
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self._concentration = array_ops.identity(
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concentration, name="concentration")
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self._rate = array_ops.identity(rate, name="rate")
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check_ops.assert_same_float_dtype(
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[self._concentration, self._rate])
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super(Gamma, self).__init__(
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dtype=self._concentration.dtype,
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validate_args=validate_args,
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allow_nan_stats=allow_nan_stats,
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reparameterization_type=distribution.FULLY_REPARAMETERIZED,
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parameters=parameters,
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graph_parents=[self._concentration,
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self._rate],
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name=name)
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@staticmethod
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def _param_shapes(sample_shape):
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return dict(
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zip(("concentration", "rate"), ([ops.convert_to_tensor(
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sample_shape, dtype=dtypes.int32)] * 2)))
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@property
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def concentration(self):
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"""Concentration parameter."""
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return self._concentration
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@property
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def rate(self):
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"""Rate parameter."""
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return self._rate
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def _batch_shape_tensor(self):
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return array_ops.broadcast_dynamic_shape(
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array_ops.shape(self.concentration),
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array_ops.shape(self.rate))
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def _batch_shape(self):
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return array_ops.broadcast_static_shape(
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self.concentration.get_shape(),
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self.rate.get_shape())
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def _event_shape_tensor(self):
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return constant_op.constant([], dtype=dtypes.int32)
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def _event_shape(self):
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return tensor_shape.scalar()
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@distribution_util.AppendDocstring(
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"""Note: See `tf.random_gamma` docstring for sampling details and
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caveats.""")
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def _sample_n(self, n, seed=None):
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return random_ops.random_gamma(
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shape=[n],
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alpha=self.concentration,
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beta=self.rate,
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dtype=self.dtype,
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seed=seed)
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def _log_prob(self, x):
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return self._log_unnormalized_prob(x) - self._log_normalization()
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def _cdf(self, x):
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x = self._maybe_assert_valid_sample(x)
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# Note that igamma returns the regularized incomplete gamma function,
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# which is what we want for the CDF.
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return math_ops.igamma(self.concentration, self.rate * x)
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def _log_unnormalized_prob(self, x):
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x = self._maybe_assert_valid_sample(x)
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return (self.concentration - 1.) * math_ops.log(x) - self.rate * x
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def _log_normalization(self):
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return (math_ops.lgamma(self.concentration)
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- self.concentration * math_ops.log(self.rate))
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def _entropy(self):
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return (self.concentration
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- math_ops.log(self.rate)
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+ math_ops.lgamma(self.concentration)
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+ ((1. - self.concentration) *
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math_ops.digamma(self.concentration)))
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def _mean(self):
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return self.concentration / self.rate
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def _variance(self):
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return self.concentration / math_ops.square(self.rate)
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def _stddev(self):
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return math_ops.sqrt(self.concentration) / self.rate
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@distribution_util.AppendDocstring(
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"""The mode of a gamma distribution is `(shape - 1) / rate` when
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`shape > 1`, and `NaN` otherwise. If `self.allow_nan_stats` is `False`,
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an exception will be raised rather than returning `NaN`.""")
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def _mode(self):
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mode = (self.concentration - 1.) / self.rate
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if self.allow_nan_stats:
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nan = array_ops.fill(
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self.batch_shape_tensor(),
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np.array(np.nan, dtype=self.dtype.as_numpy_dtype()),
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name="nan")
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return array_ops.where(self.concentration > 1., mode, nan)
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else:
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return control_flow_ops.with_dependencies([
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check_ops.assert_less(
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array_ops.ones([], self.dtype),
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self.concentration,
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message="mode not defined when any concentration <= 1"),
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], mode)
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def _maybe_assert_valid_sample(self, x):
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check_ops.assert_same_float_dtype(tensors=[x], dtype=self.dtype)
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if not self.validate_args:
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return x
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return control_flow_ops.with_dependencies([
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check_ops.assert_positive(x),
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], x)
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class GammaWithSoftplusConcentrationRate(Gamma):
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"""`Gamma` with softplus of `concentration` and `rate`."""
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def __init__(self,
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concentration,
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rate,
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validate_args=False,
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allow_nan_stats=True,
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name="GammaWithSoftplusConcentrationRate"):
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parameters = dict(locals())
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with ops.name_scope(name, values=[concentration, rate]) as name:
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super(GammaWithSoftplusConcentrationRate, self).__init__(
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concentration=nn.softplus(concentration,
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name="softplus_concentration"),
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rate=nn.softplus(rate, name="softplus_rate"),
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validate_args=validate_args,
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allow_nan_stats=allow_nan_stats,
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name=name)
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self._parameters = parameters
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@kullback_leibler.RegisterKL(Gamma, Gamma)
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def _kl_gamma_gamma(g0, g1, name=None):
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"""Calculate the batched KL divergence KL(g0 || g1) with g0 and g1 Gamma.
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Args:
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g0: instance of a Gamma distribution object.
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g1: instance of a Gamma distribution object.
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name: (optional) Name to use for created operations.
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Default is "kl_gamma_gamma".
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Returns:
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kl_gamma_gamma: `Tensor`. The batchwise KL(g0 || g1).
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"""
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with ops.name_scope(name, "kl_gamma_gamma", values=[
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g0.concentration, g0.rate, g1.concentration, g1.rate]):
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# Result from:
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# http://www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps
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# For derivation see:
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# http://stats.stackexchange.com/questions/11646/kullback-leibler-divergence-between-two-gamma-distributions pylint: disable=line-too-long
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return (((g0.concentration - g1.concentration)
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* math_ops.digamma(g0.concentration))
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+ math_ops.lgamma(g1.concentration)
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- math_ops.lgamma(g0.concentration)
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+ g1.concentration * math_ops.log(g0.rate)
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- g1.concentration * math_ops.log(g1.rate)
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+ g0.concentration * (g1.rate / g0.rate - 1.))
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