/
dirichlet_multinomial.py
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/
dirichlet_multinomial.py
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# Copyright 2018 The TensorFlow Probability Authors.
#
# 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.
# ============================================================================
"""The DirichletMultinomial distribution class."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import tensorflow.compat.v2 as tf
from tensorflow_probability.python import math as tfp_math
from tensorflow_probability.python.bijectors import softplus as softplus_bijector
from tensorflow_probability.python.distributions import distribution
from tensorflow_probability.python.distributions import gamma as gamma_lib
from tensorflow_probability.python.distributions import multinomial
from tensorflow_probability.python.internal import assert_util
from tensorflow_probability.python.internal import distribution_util
from tensorflow_probability.python.internal import dtype_util
from tensorflow_probability.python.internal import parameter_properties
from tensorflow_probability.python.internal import prefer_static as ps
from tensorflow_probability.python.internal import reparameterization
from tensorflow_probability.python.internal import samplers
from tensorflow_probability.python.internal import tensor_util
from tensorflow_probability.python.internal import tensorshape_util
__all__ = [
'DirichletMultinomial',
]
_dirichlet_multinomial_sample_note = """For each batch of counts,
`value = [n_0, ..., n_{K-1}]`, `P[value]` is the probability that after
sampling `self.total_count` draws from this Dirichlet-Multinomial distribution,
the number of draws falling in class `j` is `n_j`. Since this definition is
[exchangeable](https://en.wikipedia.org/wiki/Exchangeable_random_variables);
different sequences have the same counts so the probability includes a
combinatorial coefficient.
Note: `value` must be a non-negative tensor with dtype `self.dtype`, have no
fractional components, and such that
`tf.reduce_sum(value, -1) = self.total_count`. Its shape must be broadcastable
with `self.concentration` and `self.total_count`."""
class DirichletMultinomial(distribution.Distribution):
"""Dirichlet-Multinomial compound distribution.
The Dirichlet-Multinomial distribution is parameterized by a (batch of)
length-`K` `concentration` vectors (`K > 1`) and a `total_count` number of
trials, i.e., the number of trials per draw from the DirichletMultinomial. It
is defined over a (batch of) length-`K` vector `counts` such that
`tf.reduce_sum(counts, -1) = total_count`. The Dirichlet-Multinomial is
identically the Beta-Binomial distribution when `K = 2`.
#### Mathematical Details
The Dirichlet-Multinomial is a distribution over `K`-class counts, i.e., a
length-`K` vector of non-negative integer `counts = n = [n_0, ..., n_{K-1}]`.
The probability mass function (pmf) is,
```none
pmf(n; alpha, N) = Beta(alpha + n) / (prod_j n_j!) / Z
Z = Beta(alpha) / N!
```
where:
* `concentration = alpha = [alpha_0, ..., alpha_{K-1}]`, `alpha_j > 0`,
* `total_count = N`, `N` a positive integer,
* `N!` is `N` factorial, and,
* `Beta(x) = prod_j Gamma(x_j) / Gamma(sum_j x_j)` is 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).
Dirichlet-Multinomial is a [compound distribution](
https://en.wikipedia.org/wiki/Compound_probability_distribution), i.e., its
samples are generated as follows.
1. Choose class probabilities:
`probs = [p_0,...,p_{K-1}] ~ Dir(concentration)`
2. Draw integers:
`counts = [n_0,...,n_{K-1}] ~ Multinomial(total_count, probs)`
The last `concentration` dimension parameterizes a single
Dirichlet-Multinomial distribution. When calling distribution functions
(e.g., `dist.prob(counts)`), `concentration`, `total_count` and `counts` are
broadcast to the same shape. The last dimension of `counts` corresponds to
single Dirichlet-Multinomial distributions.
Distribution parameters are automatically broadcast in all functions; see
examples for details.
#### Pitfalls
The number of classes, `K`, must not exceed:
- the largest integer representable by `self.dtype`, i.e.,
`2**(mantissa_bits+1)` (IEE754),
- the maximum `Tensor` index, i.e., `2**31-1`.
In other words,
```python
K <= min(2**31-1, {
tf.float16: 2**11,
tf.float32: 2**24,
tf.float64: 2**53 }[param.dtype])
```
Note: This condition is validated only when `self.validate_args = True`.
#### Examples
```python
alpha = [1., 2., 3.]
n = 2.
dist = DirichletMultinomial(n, alpha)
```
Creates a 3-class distribution, with the 3rd class is most likely to be
drawn.
The distribution functions can be evaluated on counts.
```python
# counts same shape as alpha.
counts = [0., 0., 2.]
dist.prob(counts) # Shape []
# alpha will be broadcast to [[1., 2., 3.], [1., 2., 3.]] to match counts.
counts = [[1., 1., 0.], [1., 0., 1.]]
dist.prob(counts) # Shape [2]
# alpha will be broadcast to shape [5, 7, 3] to match counts.
counts = [[...]] # Shape [5, 7, 3]
dist.prob(counts) # Shape [5, 7]
```
Creates a 2-batch of 3-class distributions.
```python
alpha = [[1., 2., 3.], [4., 5., 6.]] # Shape [2, 3]
n = [3., 3.]
dist = DirichletMultinomial(n, alpha)
# counts will be broadcast to [[2., 1., 0.], [2., 1., 0.]] to match alpha.
counts = [2., 1., 0.]
dist.prob(counts) # Shape [2]
```
"""
def __init__(self,
total_count,
concentration,
validate_args=False,
allow_nan_stats=True,
name='DirichletMultinomial'):
"""Initialize a batch of DirichletMultinomial distributions.
Args:
total_count: Non-negative integer-valued tensor, whose dtype is the same
as `concentration`. The shape is broadcastable to `[N1,..., Nm]` with
`m >= 0`. Defines this as a batch of `N1 x ... x Nm` different
Dirichlet multinomial distributions. Its components should be equal to
integer values.
concentration: Positive floating point tensor with shape broadcastable to
`[N1,..., Nm, K]` `m >= 0`. Defines this as a batch of `N1 x ... x Nm`
different `K` class Dirichlet multinomial distributions.
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, 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.
"""
# Broadcasting works because:
# * The broadcasting convention is to prepend dimensions of size [1], and
# we use the last dimension for the distribution, whereas
# the batch dimensions are the leading dimensions, which forces the
# distribution dimension to be defined explicitly (i.e. it cannot be
# created automatically by prepending). This forces enough explicitness.
# * All calls involving `counts` eventually require a broadcast between
# `counts` and concentration.
# * We broadcast explicitly to include the effect of `counts` on
# `concentration` for calls that do not involve `counts`.
parameters = dict(locals())
with tf.name_scope(name) as name:
dtype = dtype_util.common_dtype([total_count, concentration], tf.float32)
self._total_count = tensor_util.convert_nonref_to_tensor(
total_count, dtype=dtype, name='total_count')
self._concentration = tensor_util.convert_nonref_to_tensor(
concentration, name='concentration')
super(DirichletMultinomial, self).__init__(
dtype=dtype,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
reparameterization_type=reparameterization.NOT_REPARAMETERIZED,
parameters=parameters,
name=name)
@classmethod
def _parameter_properties(cls, dtype, num_classes=None):
# pylint: disable=g-long-lambda
return dict(
total_count=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=parameter_properties
.BIJECTOR_NOT_IMPLEMENTED),
concentration=parameter_properties.ParameterProperties(
event_ndims=1,
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))
# pylint: enable=g-long-lambda
@property
def total_count(self):
"""Number of trials used to construct a sample."""
return self._total_count
@property
def concentration(self):
"""Concentration parameter; expected prior counts for that coordinate."""
return self._concentration
def compute_total_concentration(self):
"""Compute and return the sum of last dim of concentration parameter."""
with self._name_and_control_scope('compute_total_concentration'):
return self._compute_total_concentration()
def _compute_total_concentration(self, concentration=None):
if concentration is None:
concentration = tf.convert_to_tensor(self._concentration)
return tf.reduce_sum(concentration, axis=-1)
def _batch_shape_tensor(self, concentration=None, total_count=None):
if concentration is None:
concentration = tf.convert_to_tensor(self._concentration)
if total_count is None:
total_count = tf.convert_to_tensor(self._total_count)
return ps.broadcast_shape(
ps.shape(total_count[..., tf.newaxis]),
ps.shape(concentration))[:-1]
def _batch_shape(self):
return tensorshape_util.with_rank_at_least(
tf.broadcast_static_shape(
tf.TensorShape(self._total_count.shape).concatenate([1]),
tf.TensorShape(self._concentration.shape)),
1)[:-1]
def _event_shape_tensor(self, concentration=None):
if concentration is None:
concentration = tf.convert_to_tensor(self.concentration)
# Event shape depends only on concentration, not total_count.
return ps.shape(concentration)[-1:]
def _event_shape(self):
# Event shape depends only on concentration, not total_count.
return tensorshape_util.with_rank(self.concentration.shape[-1:], rank=1)
def _sample_n(self, n, seed=None):
gamma_seed, multinomial_seed = samplers.split_seed(
seed, salt='dirichlet_multinomial')
concentration = tf.convert_to_tensor(self._concentration)
total_count = tf.convert_to_tensor(self._total_count)
n_draws = tf.cast(total_count, dtype=tf.int32)
k = self._event_shape_tensor(concentration)[0]
alpha = tf.math.multiply(
tf.ones_like(total_count[..., tf.newaxis]),
concentration,
name='alpha')
unnormalized_logits = gamma_lib.random_gamma(
shape=[n], concentration=alpha, seed=gamma_seed, log_space=True)
x = multinomial.draw_sample(
1, k, unnormalized_logits, n_draws, self.dtype, multinomial_seed)
final_shape = ps.concat(
[[n], self._batch_shape_tensor(concentration, total_count), [k]], 0)
return tf.reshape(x, final_shape)
@distribution_util.AppendDocstring(_dirichlet_multinomial_sample_note)
def _log_prob(self, counts):
concentration = tf.convert_to_tensor(self.concentration)
ordered_prob = (
tf.math.lbeta(concentration + counts) -
tf.math.lbeta(concentration))
return ordered_prob + tfp_math.log_combinations(
self.total_count, counts)
@distribution_util.AppendDocstring(_dirichlet_multinomial_sample_note)
def _prob(self, counts):
return tf.exp(self._log_prob(counts))
def _mean(self, total_count=None, concentration=None):
if total_count is None:
total_count = tf.convert_to_tensor(self._total_count)
if concentration is None:
concentration = tf.convert_to_tensor(self._concentration)
total_concentration = self._compute_total_concentration(concentration)
scaled_concentration = (
concentration / total_concentration[..., tf.newaxis])
return total_count[..., tf.newaxis] * scaled_concentration
@distribution_util.AppendDocstring(
"""The covariance for each batch member is defined as the following:
```none
Var(X_j) = n * alpha_j / alpha_0 * (1 - alpha_j / alpha_0) *
(n + alpha_0) / (1 + alpha_0)
```
where `concentration = alpha` and
`total_concentration = alpha_0 = sum_j alpha_j`.
The covariance between elements in a batch is defined as:
```none
Cov(X_i, X_j) = -n * alpha_i * alpha_j / alpha_0 ** 2 *
(n + alpha_0) / (1 + alpha_0)
```
""")
def _covariance(self):
total_count = tf.convert_to_tensor(self._total_count)
concentration = tf.convert_to_tensor(self._concentration)
scale = self._variance_scale_term(total_count, concentration)
x = scale * self._mean(total_count, concentration)
return tf.linalg.set_diag(
-tf.matmul(x[..., tf.newaxis], x[..., tf.newaxis, :]), # outer prod
self._variance(total_count, concentration))
def _variance(self, total_count=None, concentration=None):
if total_count is None:
total_count = tf.convert_to_tensor(self._total_count)
if concentration is None:
concentration = tf.convert_to_tensor(self._concentration)
scale = self._variance_scale_term(total_count, concentration)
x = scale * self._mean(total_count, concentration)
return x * (total_count[..., tf.newaxis] * scale - x)
def _variance_scale_term(self, total_count=None, concentration=None):
"""Helper to `_covariance` and `_variance` which computes a shared scale."""
if total_count is None:
total_count = tf.convert_to_tensor(self._total_count)
if concentration is None:
concentration = tf.convert_to_tensor(self._concentration)
# Expand back the last dim so the shape of _variance_scale_term matches the
# shape of self.concentration.
c0 = self._compute_total_concentration(concentration)[..., tf.newaxis]
return tf.sqrt((1. + c0 / total_count[..., tf.newaxis]) / (1. + c0))
def _default_event_space_bijector(self):
return
def _sample_control_dependencies(self, x):
"""Checks the validity of a sample."""
assertions = []
if not self.validate_args:
return assertions
assertions.extend(distribution_util.assert_nonnegative_integer_form(x))
assertions.append(assert_util.assert_equal(
self.total_count,
tf.reduce_sum(x, axis=-1),
message='counts last-dimension must sum to `self.total_count`'))
return assertions
def _parameter_control_dependencies(self, is_init):
assertions = []
if is_init and self.validate_args:
# assert_categorical_event_shape handles both the static and dynamic case.
assertions.extend(
distribution_util.assert_categorical_event_shape(self._concentration))
if is_init != tensor_util.is_ref(self._total_count):
if self.validate_args:
total_count = tf.convert_to_tensor(self._total_count)
assertions.append(
distribution_util.assert_casting_closed(
total_count, target_dtype=tf.int32,
message='total_count cannot contain fractional components.'))
assertions.append(assert_util.assert_non_negative(
total_count, message='total_count must be non-negative'))
if is_init != tensor_util.is_ref(self._concentration):
if self.validate_args:
assertions.append(
assert_util.assert_positive(
self._concentration,
message='Concentration parameter must be positive.'))
return assertions