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tensorflow/python/ops/distributions/BUILD
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load("@xla//third_party/rules_python/python:py_library.bzl", "py_library")
package(
# copybara:uncomment default_applicable_licenses = ["//tensorflow:license"],
default_visibility = ["//tensorflow:internal"],
licenses = ["notice"],
)
py_library(
name = "distributions",
srcs = ["__init__.py"],
deprecation = ("TensorFlow Distributions has migrated to " +
"TensorFlow Probability " +
"(https://github.com/tensorflow/probability). " +
"Deprecated copies remaining in tf.distributions " +
"will not receive new features, and will be removed by " +
"early 2019. You should update all usage of " +
"`tf.distributions` to `tfp.distributions`."),
strict_deps = True,
deps = [
":distributions_py",
],
)
py_library(
name = "distributions_py",
srcs = ["distributions.py"],
strict_deps = True,
deps = [
":bernoulli",
":beta",
":categorical",
":dirichlet",
":dirichlet_multinomial",
":distribution",
":exponential",
":gamma",
":kullback_leibler",
":laplace",
":multinomial",
":normal",
":student_t",
":uniform",
"//tensorflow/python/util:deprecation",
],
)
py_library(
name = "util",
srcs = ["util.py"],
strict_deps = True,
deps = [
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/framework:tensor_util",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:array_ops_stack",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:cond",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:linalg_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/util:tf_inspect",
"//third_party/py/numpy",
],
)
py_library(
name = "kullback_leibler",
srcs = ["kullback_leibler.py"],
strict_deps = True,
deps = [
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:control_flow_assert",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//tensorflow/python/util:tf_inspect",
],
)
py_library(
name = "laplace",
srcs = ["laplace.py"],
strict_deps = True,
deps = [
":distribution",
":special_math",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//third_party/py/numpy",
],
)
py_library(
name = "dirichlet",
srcs = ["dirichlet.py"],
strict_deps = True,
deps = [
":distribution",
":kullback_leibler",
":util",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/ops:special_math_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//third_party/py/numpy",
],
)
py_library(
name = "beta",
srcs = ["beta.py"],
strict_deps = True,
deps = [
":distribution",
":kullback_leibler",
":util",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//third_party/py/numpy",
],
)
py_library(
name = "bernoulli",
srcs = ["bernoulli.py"],
strict_deps = True,
deps = [
":distribution",
":kullback_leibler",
":util",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
],
)
py_library(
name = "uniform",
srcs = ["uniform.py"],
strict_deps = True,
deps = [
":distribution",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
],
)
py_library(
name = "special_math",
srcs = ["special_math.py"],
strict_deps = True,
deps = [
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:math_ops",
"//third_party/py/numpy",
],
)
py_library(
name = "bijector_impl",
srcs = ["bijector_impl.py"],
strict_deps = True,
deps = [
":util",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/framework:tensor_util",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/util:object_identity",
"//third_party/py/numpy",
],
)
py_library(
name = "normal",
srcs = ["normal.py"],
strict_deps = True,
deps = [
":distribution",
":kullback_leibler",
":special_math",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
],
)
py_library(
name = "dirichlet_multinomial",
srcs = ["dirichlet_multinomial.py"],
strict_deps = True,
deps = [
":distribution",
":util",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/ops:special_math_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
],
)
py_library(
name = "distribution",
srcs = ["distribution.py"],
strict_deps = True,
deps = [
":kullback_leibler",
":util",
"//tensorflow/python/eager:context",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/framework:tensor_util",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//tensorflow/python/util:tf_inspect",
"//third_party/py/numpy",
],
)
py_library(
name = "identity_bijector",
srcs = ["identity_bijector.py"],
strict_deps = True,
deps = [
":bijector",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/util:deprecation",
],
)
py_library(
name = "categorical",
srcs = ["categorical.py"],
strict_deps = True,
deps = [
":distribution",
":kullback_leibler",
":util",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn_ops",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
],
)
py_library(
name = "bijector_test_util",
srcs = ["bijector_test_util.py"],
strict_deps = True,
deps = [
":uniform",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:math_ops",
"//third_party/py/numpy",
],
)
py_library(
name = "bijector",
srcs = ["bijector.py"],
strict_deps = True,
deps = [":bijector_impl"],
)
py_library(
name = "exponential",
srcs = ["exponential.py"],
strict_deps = True,
deps = [
":gamma",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//third_party/py/numpy",
],
)
py_library(
name = "gamma",
srcs = ["gamma.py"],
strict_deps = True,
deps = [
":distribution",
":kullback_leibler",
":util",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//third_party/py/numpy",
],
)
py_library(
name = "multinomial",
srcs = ["multinomial.py"],
strict_deps = True,
deps = [
":distribution",
":util",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:map_fn",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn_ops",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
],
)
py_library(
name = "transformed_distribution",
srcs = ["transformed_distribution.py"],
strict_deps = True,
deps = [
":distribution",
":identity_bijector",
":util",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/framework:tensor_util",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:array_ops_stack",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/util:deprecation",
"//third_party/py/numpy",
],
)
py_library(
name = "student_t",
srcs = ["student_t.py"],
strict_deps = True,
deps = [
":distribution",
":util",
"//tensorflow/python/framework:constant_op",
"//tensorflow/python/framework:for_generated_wrappers",
"//tensorflow/python/ops:array_ops",
"//tensorflow/python/ops:check_ops",
"//tensorflow/python/ops:control_flow_ops",
"//tensorflow/python/ops:math_ops",
"//tensorflow/python/ops:nn",
"//tensorflow/python/ops:random_ops",
"//tensorflow/python/ops:special_math_ops",
"//tensorflow/python/util:deprecation",
"//tensorflow/python/util:tf_export",
"//third_party/py/numpy",
],
)
tensorflow/python/ops/distributions/__init__.py
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# Copyright 2023 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.
# ==============================================================================
"""Core module for TensorFlow distribution objects and helpers."""
from tensorflow.python.ops.distributions import distributions
tensorflow/python/ops/distributions/bernoulli.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Bernoulli distribution class."""
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import nn
from tensorflow.python.ops import random_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
@tf_export(v1=["distributions.Bernoulli"])
class Bernoulli(distribution.Distribution):
"""Bernoulli distribution.
The Bernoulli distribution with `probs` parameter, i.e., the probability of a
`1` outcome (vs a `0` outcome).
"""
@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,
logits=None,
probs=None,
dtype=dtypes.int32,
validate_args=False,
allow_nan_stats=True,
name="Bernoulli"):
"""Construct Bernoulli distributions.
Args:
logits: An N-D `Tensor` representing the log-odds of a `1` event. Each
entry in the `Tensor` parametrizes an independent Bernoulli distribution
where the probability of an event is sigmoid(logits). Only one of
`logits` or `probs` should be passed in.
probs: An N-D `Tensor` representing the probability of a `1`
event. Each entry in the `Tensor` parameterizes an independent
Bernoulli distribution. Only one of `logits` or `probs` should be passed
in.
dtype: The type of the event samples. Default: `int32`.
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 p and logits are passed, or if neither are passed.
"""
parameters = dict(locals())
with ops.name_scope(name) as name:
self._logits, self._probs = distribution_util.get_logits_and_probs(
logits=logits,
probs=probs,
validate_args=validate_args,
name=name)
super(Bernoulli, self).__init__(
dtype=dtype,
reparameterization_type=distribution.NOT_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._logits, self._probs],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return {"logits": ops.convert_to_tensor(sample_shape, dtype=dtypes.int32)}
@property
def logits(self):
"""Log-odds of a `1` outcome (vs `0`)."""
return self._logits
@property
def probs(self):
"""Probability of a `1` outcome (vs `0`)."""
return self._probs
def _batch_shape_tensor(self):
return array_ops.shape(self._logits)
def _batch_shape(self):
return self._logits.get_shape()
def _event_shape_tensor(self):
return array_ops.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
new_shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
uniform = random_ops.random_uniform(
new_shape, seed=seed, dtype=self.probs.dtype)
sample = math_ops.less(uniform, self.probs)
return math_ops.cast(sample, self.dtype)
def _log_prob(self, event):
if self.validate_args:
event = distribution_util.embed_check_integer_casting_closed(
event, target_dtype=dtypes.bool)
# TODO(jaana): The current sigmoid_cross_entropy_with_logits has
# inconsistent behavior for logits = inf/-inf.
event = math_ops.cast(event, self.logits.dtype)
logits = self.logits
# sigmoid_cross_entropy_with_logits doesn't broadcast shape,
# so we do this here.
def _broadcast(logits, event):
return (array_ops.ones_like(event) * logits,
array_ops.ones_like(logits) * event)
if not (event.get_shape().is_fully_defined() and
logits.get_shape().is_fully_defined() and
event.get_shape() == logits.get_shape()):
logits, event = _broadcast(logits, event)
return -nn.sigmoid_cross_entropy_with_logits(labels=event, logits=logits)
def _entropy(self):
return (-self.logits * (math_ops.sigmoid(self.logits) - 1) + # pylint: disable=invalid-unary-operand-type
nn.softplus(-self.logits)) # pylint: disable=invalid-unary-operand-type
def _mean(self):
return array_ops.identity(self.probs)
def _variance(self):
return self._mean() * (1. - self.probs)
def _mode(self):
"""Returns `1` if `prob > 0.5` and `0` otherwise."""
return math_ops.cast(self.probs > 0.5, self.dtype)
@kullback_leibler.RegisterKL(Bernoulli, Bernoulli)
def _kl_bernoulli_bernoulli(a, b, name=None):
"""Calculate the batched KL divergence KL(a || b) with a and b Bernoulli.
Args:
a: instance of a Bernoulli distribution object.
b: instance of a Bernoulli distribution object.
name: (optional) Name to use for created operations.
default is "kl_bernoulli_bernoulli".
Returns:
Batchwise KL(a || b)
"""
with ops.name_scope(name, "kl_bernoulli_bernoulli",
values=[a.logits, b.logits]):
delta_probs0 = nn.softplus(-b.logits) - nn.softplus(-a.logits)
delta_probs1 = nn.softplus(b.logits) - nn.softplus(a.logits)
return (math_ops.sigmoid(a.logits) * delta_probs0
+ math_ops.sigmoid(-a.logits) * delta_probs1)
tensorflow/python/ops/distributions/beta.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Beta distribution class."""
import numpy as np
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
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 nn
from tensorflow.python.ops import random_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__ = [
"Beta",
"BetaWithSoftplusConcentration",
]
_beta_sample_note = """Note: `x` must have dtype `self.dtype` and be in
`[0, 1].` It must have a shape compatible with `self.batch_shape()`."""
@tf_export(v1=["distributions.Beta"])
class Beta(distribution.Distribution):
"""Beta distribution.
The Beta distribution is defined over the `(0, 1)` interval using parameters
`concentration1` (aka "alpha") and `concentration0` (aka "beta").
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; alpha, beta) = x**(alpha - 1) (1 - x)**(beta - 1) / Z
Z = Gamma(alpha) Gamma(beta) / Gamma(alpha + beta)
```
where:
* `concentration1 = alpha`,
* `concentration0 = beta`,
* `Z` is the normalization constant, and,
* `Gamma` is the [gamma function](
https://en.wikipedia.org/wiki/Gamma_function).
The concentration parameters represent mean total counts of a `1` or a `0`,
i.e.,
```none
concentration1 = alpha = mean * total_concentration
concentration0 = beta = (1. - mean) * total_concentration
```
where `mean` in `(0, 1)` and `total_concentration` is a positive real number
representing a mean `total_count = concentration1 + concentration0`.
Distribution parameters are automatically broadcast in all functions; see
examples for details.
Warning: 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 batch of three Beta distributions.
alpha = [1, 2, 3]
beta = [1, 2, 3]
dist = tfd.Beta(alpha, beta)
dist.sample([4, 5]) # Shape [4, 5, 3]
# `x` has three batch entries, each with two samples.
x = [[.1, .4, .5],
[.2, .3, .5]]
# Calculate the probability of each pair of samples under the corresponding
# distribution in `dist`.
dist.prob(x) # Shape [2, 3]
```
```python
# Create batch_shape=[2, 3] via parameter broadcast:
alpha = [[1.], [2]] # Shape [2, 1]
beta = [3., 4, 5] # Shape [3]
dist = tfd.Beta(alpha, beta)
# alpha broadcast as: [[1., 1, 1,],
# [2, 2, 2]]
# beta broadcast as: [[3., 4, 5],
# [3, 4, 5]]
# batch_Shape [2, 3]
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, 3]
```
Compute the gradients of samples w.r.t. the parameters:
```python
alpha = tf.constant(1.0)
beta = tf.constant(2.0)
dist = tfd.Beta(alpha, beta)
samples = dist.sample(5) # Shape [5]
loss = tf.reduce_mean(tf.square(samples)) # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, [alpha, beta])
```
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,
concentration1=None,
concentration0=None,
validate_args=False,
allow_nan_stats=True,
name="Beta"):
"""Initialize a batch of Beta distributions.
Args:
concentration1: Positive floating-point `Tensor` indicating mean
number of successes; aka "alpha". Implies `self.dtype` and
`self.batch_shape`, i.e.,
`concentration1.shape = [N1, N2, ..., Nm] = self.batch_shape`.
concentration0: Positive floating-point `Tensor` indicating mean
number of failures; aka "beta". Otherwise has same semantics as
`concentration1`.
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=[concentration1, concentration0]) as name:
self._concentration1 = self._maybe_assert_valid_concentration(
ops.convert_to_tensor(concentration1, name="concentration1"),
validate_args)
self._concentration0 = self._maybe_assert_valid_concentration(
ops.convert_to_tensor(concentration0, name="concentration0"),
validate_args)
check_ops.assert_same_float_dtype([
self._concentration1, self._concentration0])
self._total_concentration = self._concentration1 + self._concentration0
super(Beta, self).__init__(
dtype=self._total_concentration.dtype,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
reparameterization_type=distribution.FULLY_REPARAMETERIZED,
parameters=parameters,
graph_parents=[self._concentration1,
self._concentration0,
self._total_concentration],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return dict(zip(
["concentration1", "concentration0"],
[ops.convert_to_tensor(sample_shape, dtype=dtypes.int32)] * 2))
@property
def concentration1(self):
"""Concentration parameter associated with a `1` outcome."""
return self._concentration1
@property
def concentration0(self):
"""Concentration parameter associated with a `0` outcome."""
return self._concentration0
@property
def total_concentration(self):
"""Sum of concentration parameters."""
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 constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
expanded_concentration1 = array_ops.ones_like(
self.total_concentration, dtype=self.dtype) * self.concentration1
expanded_concentration0 = array_ops.ones_like(
self.total_concentration, dtype=self.dtype) * self.concentration0
gamma1_sample = random_ops.random_gamma(
shape=[n],
alpha=expanded_concentration1,
dtype=self.dtype,
seed=seed)
gamma2_sample = random_ops.random_gamma(
shape=[n],
alpha=expanded_concentration0,
dtype=self.dtype,
seed=distribution_util.gen_new_seed(seed, "beta"))
beta_sample = gamma1_sample / (gamma1_sample + gamma2_sample)
return beta_sample
@distribution_util.AppendDocstring(_beta_sample_note)
def _log_prob(self, x):
return self._log_unnormalized_prob(x) - self._log_normalization()
@distribution_util.AppendDocstring(_beta_sample_note)
def _prob(self, x):
return math_ops.exp(self._log_prob(x))
@distribution_util.AppendDocstring(_beta_sample_note)
def _log_cdf(self, x):
return math_ops.log(self._cdf(x))
@distribution_util.AppendDocstring(_beta_sample_note)
def _cdf(self, x):
return math_ops.betainc(self.concentration1, self.concentration0, x)
def _log_unnormalized_prob(self, x):
x = self._maybe_assert_valid_sample(x)
return (math_ops.xlogy(self.concentration1 - 1., x) +
(self.concentration0 - 1.) * math_ops.log1p(-x)) # pylint: disable=invalid-unary-operand-type
def _log_normalization(self):
return (math_ops.lgamma(self.concentration1)
+ math_ops.lgamma(self.concentration0)
- math_ops.lgamma(self.total_concentration))
def _entropy(self):
return (
self._log_normalization()
- (self.concentration1 - 1.) * math_ops.digamma(self.concentration1)
- (self.concentration0 - 1.) * math_ops.digamma(self.concentration0)
+ ((self.total_concentration - 2.) *
math_ops.digamma(self.total_concentration)))
def _mean(self):
return self._concentration1 / self._total_concentration
def _variance(self):
return self._mean() * (1. - self._mean()) / (1. + self.total_concentration)
@distribution_util.AppendDocstring(
"""Note: The mode is undefined when `concentration1 <= 1` or
`concentration0 <= 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):
mode = (self.concentration1 - 1.) / (self.total_concentration - 2.)
if self.allow_nan_stats:
nan = array_ops.fill(
self.batch_shape_tensor(),
np.array(np.nan, dtype=self.dtype.as_numpy_dtype()),
name="nan")
is_defined = math_ops.logical_and(self.concentration1 > 1.,
self.concentration0 > 1.)
return array_ops.where_v2(is_defined, mode, nan)
return control_flow_ops.with_dependencies([
check_ops.assert_less(
array_ops.ones([], dtype=self.dtype),
self.concentration1,
message="Mode undefined for concentration1 <= 1."),
check_ops.assert_less(
array_ops.ones([], dtype=self.dtype),
self.concentration0,
message="Mode undefined for concentration0 <= 1.")
], mode)
def _maybe_assert_valid_concentration(self, concentration, validate_args):
"""Checks the validity of a 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."),
], 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="sample must be positive"),
check_ops.assert_less(
x,
array_ops.ones([], self.dtype),
message="sample must be less than `1`."),
], x)
class BetaWithSoftplusConcentration(Beta):
"""Beta with softplus transform of `concentration1` and `concentration0`."""
@deprecation.deprecated(
"2019-01-01",
"Use `tfd.Beta(tf.nn.softplus(concentration1), "
"tf.nn.softplus(concentration2))` instead.",
warn_once=True)
def __init__(self,
concentration1,
concentration0,
validate_args=False,
allow_nan_stats=True,
name="BetaWithSoftplusConcentration"):
parameters = dict(locals())
with ops.name_scope(name, values=[concentration1,
concentration0]) as name:
super(BetaWithSoftplusConcentration, self).__init__(
concentration1=nn.softplus(concentration1,
name="softplus_concentration1"),
concentration0=nn.softplus(concentration0,
name="softplus_concentration0"),
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
@kullback_leibler.RegisterKL(Beta, Beta)
def _kl_beta_beta(d1, d2, name=None):
"""Calculate the batchwise KL divergence KL(d1 || d2) with d1 and d2 Beta.
Args:
d1: instance of a Beta distribution object.
d2: instance of a Beta distribution object.
name: (optional) Name to use for created operations.
default is "kl_beta_beta".
Returns:
Batchwise KL(d1 || d2)
"""
def delta(fn, is_property=True):
fn1 = getattr(d1, fn)
fn2 = getattr(d2, fn)
return (fn2 - fn1) if is_property else (fn2() - fn1())
with ops.name_scope(name, "kl_beta_beta", values=[
d1.concentration1,
d1.concentration0,
d1.total_concentration,
d2.concentration1,
d2.concentration0,
d2.total_concentration,
]):
return (delta("_log_normalization", is_property=False)
- math_ops.digamma(d1.concentration1) * delta("concentration1")
- math_ops.digamma(d1.concentration0) * delta("concentration0")
+ (math_ops.digamma(d1.total_concentration)
* delta("total_concentration")))
tensorflow/python/ops/distributions/bijector.py
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# 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
# limitations under the License.
# ==============================================================================
"""Bijector base."""
# go/tf-wildcard-import
# pylint: disable=wildcard-import,unused-import
from tensorflow.python.ops.distributions.bijector_impl import Bijector
# pylint: enable=wildcard-import,unused-import
tensorflow/python/ops/distributions/bijector_impl.py
+1113
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tensorflow/python/ops/distributions/bijector_test_util.py
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# 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
# limitations under the License.
# ==============================================================================
"""Bijector unit-test utilities."""
import numpy as np
from tensorflow.python.framework import ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops.distributions import uniform as uniform_lib
def assert_finite(array):
if not np.isfinite(array).all():
raise AssertionError("array was not all finite. %s" % array[:15])
def assert_strictly_increasing(array):
np.testing.assert_array_less(0., np.diff(array))
def assert_strictly_decreasing(array):
np.testing.assert_array_less(np.diff(array), 0.)
def assert_strictly_monotonic(array):
if array[0] < array[-1]:
assert_strictly_increasing(array)
else:
assert_strictly_decreasing(array)
def assert_scalar_congruency(bijector,
lower_x,
upper_x,
n=int(10e3),
rtol=0.01,
sess=None):
"""Assert `bijector`'s forward/inverse/inverse_log_det_jacobian are congruent.
We draw samples `X ~ U(lower_x, upper_x)`, then feed these through the
`bijector` in order to check that:
1. the forward is strictly monotonic.
2. the forward/inverse methods are inverses of each other.
3. the jacobian is the correct change of measure.
This can only be used for a Bijector mapping open subsets of the real line
to themselves. This is due to the fact that this test compares the `prob`
before/after transformation with the Lebesgue measure on the line.
Args:
bijector: Instance of Bijector
lower_x: Python scalar.
upper_x: Python scalar. Must have `lower_x < upper_x`, and both must be in
the domain of the `bijector`. The `bijector` should probably not produce
huge variation in values in the interval `(lower_x, upper_x)`, or else
the variance based check of the Jacobian will require small `rtol` or
huge `n`.
n: Number of samples to draw for the checks.
rtol: Positive number. Used for the Jacobian check.
sess: `tf.compat.v1.Session`. Defaults to the default session.
Raises:
AssertionError: If tests fail.
"""
# Checks and defaults.
if sess is None:
sess = ops.get_default_session()
# Should be monotonic over this interval
ten_x_pts = np.linspace(lower_x, upper_x, num=10).astype(np.float32)
if bijector.dtype is not None:
ten_x_pts = ten_x_pts.astype(bijector.dtype.as_numpy_dtype)
forward_on_10_pts = bijector.forward(ten_x_pts)
# Set the lower/upper limits in the range of the bijector.
lower_y, upper_y = sess.run(
[bijector.forward(lower_x), bijector.forward(upper_x)])
if upper_y < lower_y: # If bijector.forward is a decreasing function.
lower_y, upper_y = upper_y, lower_y
# Uniform samples from the domain, range.
uniform_x_samps = uniform_lib.Uniform(
low=lower_x, high=upper_x).sample(n, seed=0)
uniform_y_samps = uniform_lib.Uniform(
low=lower_y, high=upper_y).sample(n, seed=1)
# These compositions should be the identity.
inverse_forward_x = bijector.inverse(bijector.forward(uniform_x_samps))
forward_inverse_y = bijector.forward(bijector.inverse(uniform_y_samps))
# For a < b, and transformation y = y(x),
# (b - a) = \int_a^b dx = \int_{y(a)}^{y(b)} |dx/dy| dy
# "change_measure_dy_dx" below is a Monte Carlo approximation to the right
# hand side, which should then be close to the left, which is (b - a).
# We assume event_ndims=0 because we assume scalar -> scalar. The log_det
# methods will handle whether they expect event_ndims > 0.
dy_dx = math_ops.exp(bijector.inverse_log_det_jacobian(
uniform_y_samps, event_ndims=0))
# E[|dx/dy|] under Uniform[lower_y, upper_y]
# = \int_{y(a)}^{y(b)} |dx/dy| dP(u), where dP(u) is the uniform measure
expectation_of_dy_dx_under_uniform = math_ops.reduce_mean(dy_dx)
# dy = dP(u) * (upper_y - lower_y)
change_measure_dy_dx = (
(upper_y - lower_y) * expectation_of_dy_dx_under_uniform)
# We'll also check that dy_dx = 1 / dx_dy.
dx_dy = math_ops.exp(
bijector.forward_log_det_jacobian(
bijector.inverse(uniform_y_samps), event_ndims=0))
[
forward_on_10_pts_v,
dy_dx_v,
dx_dy_v,
change_measure_dy_dx_v,
uniform_x_samps_v,
uniform_y_samps_v,
inverse_forward_x_v,
forward_inverse_y_v,
] = sess.run([
forward_on_10_pts,
dy_dx,
dx_dy,
change_measure_dy_dx,
uniform_x_samps,
uniform_y_samps,
inverse_forward_x,
forward_inverse_y,
])
assert_strictly_monotonic(forward_on_10_pts_v)
# Composition of forward/inverse should be the identity.
np.testing.assert_allclose(
inverse_forward_x_v, uniform_x_samps_v, atol=1e-5, rtol=1e-3)
np.testing.assert_allclose(
forward_inverse_y_v, uniform_y_samps_v, atol=1e-5, rtol=1e-3)
# Change of measure should be correct.
np.testing.assert_allclose(
upper_x - lower_x, change_measure_dy_dx_v, atol=0, rtol=rtol)
# Inverse Jacobian should be equivalent to the reciprocal of the forward
# Jacobian.
np.testing.assert_allclose(
dy_dx_v, np.divide(1., dx_dy_v), atol=1e-5, rtol=1e-3)
def assert_bijective_and_finite(
bijector, x, y, event_ndims, atol=0, rtol=1e-5, sess=None):
"""Assert that forward/inverse (along with jacobians) are inverses and finite.
It is recommended to use x and y values that are very very close to the edge
of the Bijector's domain.
Args:
bijector: A Bijector instance.
x: np.array of values in the domain of bijector.forward.
y: np.array of values in the domain of bijector.inverse.
event_ndims: Integer describing the number of event dimensions this bijector
operates on.
atol: Absolute tolerance.
rtol: Relative tolerance.
sess: TensorFlow session. Defaults to the default session.
Raises:
AssertionError: If tests fail.
"""
sess = sess or ops.get_default_session()
# These are the incoming points, but people often create a crazy range of
# values for which these end up being bad, especially in 16bit.
assert_finite(x)
assert_finite(y)
f_x = bijector.forward(x)
g_y = bijector.inverse(y)
[
x_from_x,
y_from_y,
ildj_f_x,
fldj_x,
ildj_y,
fldj_g_y,
f_x_v,
g_y_v,
] = sess.run([
bijector.inverse(f_x),
bijector.forward(g_y),
bijector.inverse_log_det_jacobian(f_x, event_ndims=event_ndims),
bijector.forward_log_det_jacobian(x, event_ndims=event_ndims),
bijector.inverse_log_det_jacobian(y, event_ndims=event_ndims),
bijector.forward_log_det_jacobian(g_y, event_ndims=event_ndims),
f_x,
g_y,
])
assert_finite(x_from_x)
assert_finite(y_from_y)
assert_finite(ildj_f_x)
assert_finite(fldj_x)
assert_finite(ildj_y)
assert_finite(fldj_g_y)
assert_finite(f_x_v)
assert_finite(g_y_v)
np.testing.assert_allclose(x_from_x, x, atol=atol, rtol=rtol)
np.testing.assert_allclose(y_from_y, y, atol=atol, rtol=rtol)
np.testing.assert_allclose(-ildj_f_x, fldj_x, atol=atol, rtol=rtol)
np.testing.assert_allclose(-ildj_y, fldj_g_y, atol=atol, rtol=rtol)
tensorflow/python/ops/distributions/categorical.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Categorical distribution class."""
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import nn_ops
from tensorflow.python.ops import random_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
def _broadcast_cat_event_and_params(event, params, base_dtype):
"""Broadcasts the event or distribution parameters."""
if event.dtype.is_integer:
pass
elif event.dtype.is_floating:
# When `validate_args=True` we've already ensured int/float casting
# is closed.
event = math_ops.cast(event, dtype=dtypes.int32)
else:
raise TypeError("`value` should have integer `dtype` or "
"`self.dtype` ({})".format(base_dtype))
shape_known_statically = (
params.shape.ndims is not None and
params.shape[:-1].is_fully_defined() and
event.shape.is_fully_defined())
if not shape_known_statically or params.shape[:-1] != event.shape:
params *= array_ops.ones_like(event[..., array_ops.newaxis],
dtype=params.dtype)
params_shape = array_ops.shape(params)[:-1]
event *= array_ops.ones(params_shape, dtype=event.dtype)
if params.shape.ndims is not None:
event.set_shape(tensor_shape.TensorShape(params.shape[:-1]))
return event, params
@tf_export(v1=["distributions.Categorical"])
class Categorical(distribution.Distribution):
"""Categorical distribution.
The Categorical distribution is parameterized by either probabilities or
log-probabilities of a set of `K` classes. It is defined over the integers
`{0, 1, ..., K}`.
The Categorical distribution is closely related to the `OneHotCategorical` and
`Multinomial` distributions. The Categorical distribution can be intuited as
generating samples according to `argmax{ OneHotCategorical(probs) }` itself
being identical to `argmax{ Multinomial(probs, total_count=1) }`.
#### Mathematical Details
The probability mass function (pmf) is,
```none
pmf(k; pi) = prod_j pi_j**[k == j]
```
#### Pitfalls
The number of classes, `K`, must not exceed:
- the largest integer representable by `self.dtype`, i.e.,
`2**(mantissa_bits+1)` (IEEE 754),
- 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
Creates a 3-class distribution with the 2nd class being most likely.
```python
dist = Categorical(probs=[0.1, 0.5, 0.4])
n = 1e4
empirical_prob = tf.cast(
tf.histogram_fixed_width(
dist.sample(int(n)),
[0., 2],
nbins=3),
dtype=tf.float32) / n
# ==> array([ 0.1005, 0.5037, 0.3958], dtype=float32)
```
Creates a 3-class distribution with the 2nd class being most likely.
Parameterized by [logits](https://en.wikipedia.org/wiki/Logit) rather than
probabilities.
```python
dist = Categorical(logits=np.log([0.1, 0.5, 0.4])
n = 1e4
empirical_prob = tf.cast(
tf.histogram_fixed_width(
dist.sample(int(n)),
[0., 2],
nbins=3),
dtype=tf.float32) / n
# ==> array([0.1045, 0.5047, 0.3908], dtype=float32)
```
Creates a 3-class distribution with the 3rd class being most likely.
The distribution functions can be evaluated on counts.
```python
# counts is a scalar.
p = [0.1, 0.4, 0.5]
dist = Categorical(probs=p)
dist.prob(0) # Shape []
# p will be broadcast to [[0.1, 0.4, 0.5], [0.1, 0.4, 0.5]] to match counts.
counts = [1, 0]
dist.prob(counts) # Shape [2]
# p will be broadcast to shape [3, 5, 7, 3] to match counts.
counts = [[...]] # Shape [5, 7, 3]
dist.prob(counts) # Shape [5, 7, 3]
```
"""
@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,
logits=None,
probs=None,
dtype=dtypes.int32,
validate_args=False,
allow_nan_stats=True,
name="Categorical"):
"""Initialize Categorical distributions using class log-probabilities.
Args:
logits: An N-D `Tensor`, `N >= 1`, representing the log probabilities
of a set of Categorical distributions. The first `N - 1` dimensions
index into a batch of independent distributions and the last dimension
represents a vector of logits for each class. Only one of `logits` or
`probs` should be passed in.
probs: An N-D `Tensor`, `N >= 1`, representing the probabilities
of a set of Categorical distributions. The first `N - 1` dimensions
index into a batch of independent distributions and the last dimension
represents a vector of probabilities for each class. Only one of
`logits` or `probs` should be passed in.
dtype: The type of the event samples (default: int32).
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=[logits, probs]) as name:
self._logits, self._probs = distribution_util.get_logits_and_probs(
logits=logits,
probs=probs,
validate_args=validate_args,
multidimensional=True,
name=name)
if validate_args:
self._logits = distribution_util.embed_check_categorical_event_shape(
self._logits)
logits_shape_static = self._logits.get_shape().with_rank_at_least(1)
if logits_shape_static.ndims is not None:
self._batch_rank = ops.convert_to_tensor(
logits_shape_static.ndims - 1,
dtype=dtypes.int32,
name="batch_rank")
else:
with ops.name_scope(name="batch_rank"):
self._batch_rank = array_ops.rank(self._logits) - 1
logits_shape = array_ops.shape(self._logits, name="logits_shape")
if tensor_shape.dimension_value(logits_shape_static[-1]) is not None:
self._event_size = ops.convert_to_tensor(
logits_shape_static.dims[-1].value,
dtype=dtypes.int32,
name="event_size")
else:
with ops.name_scope(name="event_size"):
self._event_size = logits_shape[self._batch_rank]
if logits_shape_static[:-1].is_fully_defined():
self._batch_shape_val = constant_op.constant(
logits_shape_static[:-1].as_list(),
dtype=dtypes.int32,
name="batch_shape")
else:
with ops.name_scope(name="batch_shape"):
self._batch_shape_val = logits_shape[:-1]
super(Categorical, self).__init__(
dtype=dtype,
reparameterization_type=distribution.NOT_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._logits,
self._probs],
name=name)
@property
def event_size(self):
"""Scalar `int32` tensor: the number of classes."""
return self._event_size
@property
def logits(self):
"""Vector of coordinatewise logits."""
return self._logits
@property
def probs(self):
"""Vector of coordinatewise probabilities."""
return self._probs
def _batch_shape_tensor(self):
return array_ops.identity(self._batch_shape_val)
def _batch_shape(self):
return self.logits.get_shape()[:-1]
def _event_shape_tensor(self):
return constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
if self.logits.get_shape().ndims == 2:
logits_2d = self.logits
else:
logits_2d = array_ops.reshape(self.logits, [-1, self.event_size])
sample_dtype = dtypes.int64 if self.dtype.size > 4 else dtypes.int32
draws = random_ops.multinomial(
logits_2d, n, seed=seed, output_dtype=sample_dtype)
draws = array_ops.reshape(
array_ops.transpose(draws),
array_ops.concat([[n], self.batch_shape_tensor()], 0))
return math_ops.cast(draws, self.dtype)
def _cdf(self, k):
k = ops.convert_to_tensor(k, name="k")
if self.validate_args:
k = distribution_util.embed_check_integer_casting_closed(
k, target_dtype=dtypes.int32)
k, probs = _broadcast_cat_event_and_params(
k, self.probs, base_dtype=self.dtype.base_dtype)
# batch-flatten everything in order to use `sequence_mask()`.
batch_flattened_probs = array_ops.reshape(probs,
(-1, self._event_size))
batch_flattened_k = array_ops.reshape(k, [-1])
to_sum_over = array_ops.where(
array_ops.sequence_mask(batch_flattened_k, self._event_size),
batch_flattened_probs,
array_ops.zeros_like(batch_flattened_probs))
batch_flattened_cdf = math_ops.reduce_sum(to_sum_over, axis=-1)
# Reshape back to the shape of the argument.
return array_ops.reshape(batch_flattened_cdf, array_ops.shape(k))
def _log_prob(self, k):
k = ops.convert_to_tensor(k, name="k")
if self.validate_args:
k = distribution_util.embed_check_integer_casting_closed(
k, target_dtype=dtypes.int32)
k, logits = _broadcast_cat_event_and_params(
k, self.logits, base_dtype=self.dtype.base_dtype)
# pylint: disable=invalid-unary-operand-type
return -nn_ops.sparse_softmax_cross_entropy_with_logits(
labels=k,
logits=logits)
def _entropy(self):
return -math_ops.reduce_sum(
nn_ops.log_softmax(self.logits) * self.probs, axis=-1)
def _mode(self):
ret = math_ops.argmax(self.logits, axis=self._batch_rank)
ret = math_ops.cast(ret, self.dtype)
ret.set_shape(self.batch_shape)
return ret
@kullback_leibler.RegisterKL(Categorical, Categorical)
def _kl_categorical_categorical(a, b, name=None):
"""Calculate the batched KL divergence KL(a || b) with a and b Categorical.
Args:
a: instance of a Categorical distribution object.
b: instance of a Categorical distribution object.
name: (optional) Name to use for created operations.
default is "kl_categorical_categorical".
Returns:
Batchwise KL(a || b)
"""
with ops.name_scope(name, "kl_categorical_categorical",
values=[a.logits, b.logits]):
# sum(probs log(probs / (1 - probs)))
delta_log_probs1 = (nn_ops.log_softmax(a.logits) -
nn_ops.log_softmax(b.logits))
return math_ops.reduce_sum(nn_ops.softmax(a.logits) * delta_log_probs1,
axis=-1)
tensorflow/python/ops/distributions/dirichlet.py
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# 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
# limitations under the License.
# ==============================================================================
"""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))
tensorflow/python/ops/distributions/dirichlet_multinomial.py
+353
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# 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
# limitations under the License.
# ==============================================================================
"""The DirichletMultinomial distribution class."""
from tensorflow.python.framework import dtypes
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 util as distribution_util
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export
__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`."""
@tf_export(v1=["distributions.DirichletMultinomial"])
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 parametrizes 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 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]
```
"""
# TODO(b/27419586) Change docstring for dtype of concentration once int
# allowed.
@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,
total_count,
concentration,
validate_args=False,
allow_nan_stats=True,
name="DirichletMultinomial"):
"""Initialize a batch of DirichletMultinomial distributions.
Args:
total_count: Non-negative floating point 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, whose dtype is the
same as `n` 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, 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=[total_count, concentration]) as name:
# 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.
self._total_count = ops.convert_to_tensor(total_count, name="total_count")
if validate_args:
self._total_count = (
distribution_util.embed_check_nonnegative_integer_form(
self._total_count))
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(DirichletMultinomial, self).__init__(
dtype=self._concentration.dtype,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
reparameterization_type=distribution.NOT_REPARAMETERIZED,
parameters=parameters,
graph_parents=[self._total_count,
self._concentration],
name=name)
@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
@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):
# Event shape depends only on total_concentration, not "n".
return self.concentration.get_shape().with_rank_at_least(1)[-1:]
def _sample_n(self, n, seed=None):
n_draws = math_ops.cast(self.total_count, dtype=dtypes.int32)
k = self.event_shape_tensor()[0]
unnormalized_logits = array_ops.reshape(
math_ops.log(random_ops.random_gamma(
shape=[n],
alpha=self.concentration,
dtype=self.dtype,
seed=seed)),
shape=[-1, k])
draws = random_ops.multinomial(
logits=unnormalized_logits,
num_samples=n_draws,
seed=distribution_util.gen_new_seed(seed, salt="dirichlet_multinomial"))
x = math_ops.reduce_sum(array_ops.one_hot(draws, depth=k), -2)
final_shape = array_ops.concat([[n], self.batch_shape_tensor(), [k]], 0)
x = array_ops.reshape(x, final_shape)
return math_ops.cast(x, self.dtype)
@distribution_util.AppendDocstring(_dirichlet_multinomial_sample_note)
def _log_prob(self, counts):
counts = self._maybe_assert_valid_sample(counts)
ordered_prob = (
special_math_ops.lbeta(self.concentration + counts)
- special_math_ops.lbeta(self.concentration))
return ordered_prob + distribution_util.log_combinations(
self.total_count, counts)
@distribution_util.AppendDocstring(_dirichlet_multinomial_sample_note)
def _prob(self, counts):
return math_ops.exp(self._log_prob(counts))
def _mean(self):
return self.total_count * (self.concentration /
self.total_concentration[..., array_ops.newaxis])
@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):
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 * (self.total_count * scale - x)
def _variance_scale_term(self):
"""Helper to `_covariance` and `_variance` which computes a shared scale."""
# We must take care to expand back the last dim whenever we use the
# total_concentration.
c0 = self.total_concentration[..., array_ops.newaxis]
return math_ops.sqrt((1. + c0 / self.total_count) / (1. + c0))
def _maybe_assert_valid_concentration(self, concentration, validate_args):
"""Checks the validity of the concentration parameter."""
if not validate_args:
return concentration
concentration = distribution_util.embed_check_categorical_event_shape(
concentration)
return control_flow_ops.with_dependencies([
check_ops.assert_positive(
concentration,
message="Concentration parameter must be positive."),
], concentration)
def _maybe_assert_valid_sample(self, counts):
"""Check counts for proper shape, values, then return tensor version."""
if not self.validate_args:
return counts
counts = distribution_util.embed_check_nonnegative_integer_form(counts)
return control_flow_ops.with_dependencies([
check_ops.assert_equal(
self.total_count, math_ops.reduce_sum(counts, -1),
message="counts last-dimension must sum to `self.total_count`"),
], counts)
tensorflow/python/ops/distributions/distribution.py
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tensorflow/python/ops/distributions/distributions.py
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# 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.
# ==============================================================================
"""Core module for TensorFlow distribution objects and helpers."""
from tensorflow.python.util import deprecation
# pylint: disable=wildcard-import,unused-import,g-import-not-at-top
with deprecation.silence():
from tensorflow.python.ops.distributions.bernoulli import Bernoulli
from tensorflow.python.ops.distributions.beta import Beta
from tensorflow.python.ops.distributions.categorical import Categorical
from tensorflow.python.ops.distributions.dirichlet import Dirichlet
from tensorflow.python.ops.distributions.dirichlet_multinomial import DirichletMultinomial
from tensorflow.python.ops.distributions.distribution import *
from tensorflow.python.ops.distributions.exponential import Exponential
from tensorflow.python.ops.distributions.gamma import Gamma
from tensorflow.python.ops.distributions.kullback_leibler import *
from tensorflow.python.ops.distributions.laplace import Laplace
from tensorflow.python.ops.distributions.multinomial import Multinomial
from tensorflow.python.ops.distributions.normal import Normal
from tensorflow.python.ops.distributions.student_t import StudentT
from tensorflow.python.ops.distributions.uniform import Uniform
# pylint: enable=wildcard-import,unused-import
del deprecation
tensorflow/python/ops/distributions/exponential.py
+162
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# 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
# limitations under the License.
# ==============================================================================
"""The Exponential distribution class."""
import numpy as np
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import nn
from tensorflow.python.ops import random_ops
from tensorflow.python.ops.distributions import gamma
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export
__all__ = [
"Exponential",
"ExponentialWithSoftplusRate",
]
@tf_export(v1=["distributions.Exponential"])
class Exponential(gamma.Gamma):
"""Exponential distribution.
The Exponential distribution is parameterized by an event `rate` parameter.
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; lambda, x > 0) = exp(-lambda x) / Z
Z = 1 / lambda
```
where `rate = lambda` and `Z` is the normalizaing constant.
The Exponential distribution is a special case of the Gamma distribution,
i.e.,
```python
Exponential(rate) = Gamma(concentration=1., rate)
```
The Exponential distribution uses a `rate` parameter, or "inverse scale",
which can be intuited as,
```none
X ~ Exponential(rate=1)
Y = X / rate
```
"""
@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,
rate,
validate_args=False,
allow_nan_stats=True,
name="Exponential"):
"""Construct Exponential distribution with parameter `rate`.
Args:
rate: Floating point tensor, equivalent to `1 / mean`. Must contain only
positive values.
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())
# Even though all statistics of are defined for valid inputs, this is not
# true in the parent class "Gamma." Therefore, passing
# allow_nan_stats=True
# through to the parent class results in unnecessary asserts.
with ops.name_scope(name, values=[rate]) as name:
self._rate = ops.convert_to_tensor(rate, name="rate")
super(Exponential, self).__init__(
concentration=array_ops.ones([], dtype=self._rate.dtype),
rate=self._rate,
allow_nan_stats=allow_nan_stats,
validate_args=validate_args,
name=name)
self._parameters = parameters
self._graph_parents += [self._rate]
@staticmethod
def _param_shapes(sample_shape):
return {"rate": ops.convert_to_tensor(sample_shape, dtype=dtypes.int32)}
@property
def rate(self):
return self._rate
def _log_survival_function(self, value):
return self._log_prob(value) - math_ops.log(self._rate)
def _sample_n(self, n, seed=None):
shape = array_ops.concat([[n], array_ops.shape(self._rate)], 0)
# Uniform variates must be sampled from the open-interval `(0, 1)` rather
# than `[0, 1)`. To do so, we use `np.finfo(self.dtype.as_numpy_dtype).tiny`
# because it is the smallest, positive, "normal" number. A "normal" number
# is such that the mantissa has an implicit leading 1. Normal, positive
# numbers x, y have the reasonable property that, `x + y >= max(x, y)`. In
# this case, a subnormal number (i.e., np.nextafter) can cause us to sample
# 0.
sampled = random_ops.random_uniform(
shape,
minval=np.finfo(self.dtype.as_numpy_dtype).tiny,
maxval=1.,
seed=seed,
dtype=self.dtype)
return -math_ops.log(sampled) / self._rate
class ExponentialWithSoftplusRate(Exponential):
"""Exponential with softplus transform on `rate`."""
@deprecation.deprecated(
"2019-01-01",
"Use `tfd.Exponential(tf.nn.softplus(rate)).",
warn_once=True)
def __init__(self,
rate,
validate_args=False,
allow_nan_stats=True,
name="ExponentialWithSoftplusRate"):
parameters = dict(locals())
with ops.name_scope(name, values=[rate]) as name:
super(ExponentialWithSoftplusRate, self).__init__(
rate=nn.softplus(rate, name="softplus_rate"),
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
tensorflow/python/ops/distributions/gamma.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Gamma distribution class."""
import numpy as np
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
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 nn
from tensorflow.python.ops import random_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__ = [
"Gamma",
"GammaWithSoftplusConcentrationRate",
]
@tf_export(v1=["distributions.Gamma"])
class Gamma(distribution.Distribution):
"""Gamma distribution.
The Gamma distribution is defined over positive real numbers using
parameters `concentration` (aka "alpha") and `rate` (aka "beta").
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; alpha, beta, x > 0) = x**(alpha - 1) exp(-x beta) / Z
Z = Gamma(alpha) beta**(-alpha)
```
where:
* `concentration = alpha`, `alpha > 0`,
* `rate = beta`, `beta > 0`,
* `Z` is the normalizing constant, and,
* `Gamma` is the [gamma function](
https://en.wikipedia.org/wiki/Gamma_function).
The cumulative density function (cdf) is,
```none
cdf(x; alpha, beta, x > 0) = GammaInc(alpha, beta x) / Gamma(alpha)
```
where `GammaInc` is the [lower incomplete Gamma function](
https://en.wikipedia.org/wiki/Incomplete_gamma_function).
The parameters can be intuited via their relationship to mean and stddev,
```none
concentration = alpha = (mean / stddev)**2
rate = beta = mean / stddev**2 = concentration / mean
```
Distribution parameters are automatically broadcast in all functions; see
examples for details.
Warning: The samples of this distribution are always non-negative. However,
the samples that are smaller than `np.finfo(dtype).tiny` are rounded
to this value, so it appears more often than it should.
This should only be noticeable when the `concentration` is very small, or the
`rate` is very large. See note in `tf.random.gamma` docstring.
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
dist = tfd.Gamma(concentration=3.0, rate=2.0)
dist2 = tfd.Gamma(concentration=[3.0, 4.0], rate=[2.0, 3.0])
```
Compute the gradients of samples w.r.t. the parameters:
```python
concentration = tf.constant(3.0)
rate = tf.constant(2.0)
dist = tfd.Gamma(concentration, rate)
samples = dist.sample(5) # Shape [5]
loss = tf.reduce_mean(tf.square(samples)) # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, [concentration, rate])
```
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,
rate,
validate_args=False,
allow_nan_stats=True,
name="Gamma"):
"""Construct Gamma with `concentration` and `rate` parameters.
The parameters `concentration` and `rate` must be shaped in a way that
supports broadcasting (e.g. `concentration + rate` is a valid operation).
Args:
concentration: Floating point tensor, the concentration params of the
distribution(s). Must contain only positive values.
rate: Floating point tensor, the inverse scale params of the
distribution(s). Must contain only positive values.
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:
TypeError: if `concentration` and `rate` are different dtypes.
"""
parameters = dict(locals())
with ops.name_scope(name, values=[concentration, rate]) as name:
with ops.control_dependencies([
check_ops.assert_positive(concentration),
check_ops.assert_positive(rate),
] if validate_args else []):
self._concentration = array_ops.identity(
concentration, name="concentration")
self._rate = array_ops.identity(rate, name="rate")
check_ops.assert_same_float_dtype(
[self._concentration, self._rate])
super(Gamma, 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._rate],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return dict(
zip(("concentration", "rate"), ([ops.convert_to_tensor(
sample_shape, dtype=dtypes.int32)] * 2)))
@property
def concentration(self):
"""Concentration parameter."""
return self._concentration
@property
def rate(self):
"""Rate parameter."""
return self._rate
def _batch_shape_tensor(self):
return array_ops.broadcast_dynamic_shape(
array_ops.shape(self.concentration),
array_ops.shape(self.rate))
def _batch_shape(self):
return array_ops.broadcast_static_shape(
self.concentration.get_shape(),
self.rate.get_shape())
def _event_shape_tensor(self):
return constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
@distribution_util.AppendDocstring(
"""Note: See `tf.random.gamma` docstring for sampling details and
caveats.""")
def _sample_n(self, n, seed=None):
return random_ops.random_gamma(
shape=[n],
alpha=self.concentration,
beta=self.rate,
dtype=self.dtype,
seed=seed)
def _log_prob(self, x):
return self._log_unnormalized_prob(x) - self._log_normalization()
def _cdf(self, x):
x = self._maybe_assert_valid_sample(x)
# Note that igamma returns the regularized incomplete gamma function,
# which is what we want for the CDF.
return math_ops.igamma(self.concentration, self.rate * x)
def _log_unnormalized_prob(self, x):
x = self._maybe_assert_valid_sample(x)
return math_ops.xlogy(self.concentration - 1., x) - self.rate * x
def _log_normalization(self):
return (math_ops.lgamma(self.concentration)
- self.concentration * math_ops.log(self.rate))
def _entropy(self):
return (self.concentration
- math_ops.log(self.rate)
+ math_ops.lgamma(self.concentration)
+ ((1. - self.concentration) *
math_ops.digamma(self.concentration)))
def _mean(self):
return self.concentration / self.rate
def _variance(self):
return self.concentration / math_ops.square(self.rate)
def _stddev(self):
return math_ops.sqrt(self.concentration) / self.rate
@distribution_util.AppendDocstring(
"""The mode of a gamma distribution is `(shape - 1) / rate` when
`shape > 1`, and `NaN` otherwise. If `self.allow_nan_stats` is `False`,
an exception will be raised rather than returning `NaN`.""")
def _mode(self):
mode = (self.concentration - 1.) / self.rate
if self.allow_nan_stats:
nan = array_ops.fill(
self.batch_shape_tensor(),
np.array(np.nan, dtype=self.dtype.as_numpy_dtype()),
name="nan")
return array_ops.where_v2(self.concentration > 1., mode, nan)
else:
return control_flow_ops.with_dependencies([
check_ops.assert_less(
array_ops.ones([], self.dtype),
self.concentration,
message="mode not defined when any concentration <= 1"),
], mode)
def _maybe_assert_valid_sample(self, x):
check_ops.assert_same_float_dtype(tensors=[x], dtype=self.dtype)
if not self.validate_args:
return x
return control_flow_ops.with_dependencies([
check_ops.assert_positive(x),
], x)
class GammaWithSoftplusConcentrationRate(Gamma):
"""`Gamma` with softplus of `concentration` and `rate`."""
@deprecation.deprecated(
"2019-01-01",
"Use `tfd.Gamma(tf.nn.softplus(concentration), "
"tf.nn.softplus(rate))` instead.",
warn_once=True)
def __init__(self,
concentration,
rate,
validate_args=False,
allow_nan_stats=True,
name="GammaWithSoftplusConcentrationRate"):
parameters = dict(locals())
with ops.name_scope(name, values=[concentration, rate]) as name:
super(GammaWithSoftplusConcentrationRate, self).__init__(
concentration=nn.softplus(concentration,
name="softplus_concentration"),
rate=nn.softplus(rate, name="softplus_rate"),
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
@kullback_leibler.RegisterKL(Gamma, Gamma)
def _kl_gamma_gamma(g0, g1, name=None):
"""Calculate the batched KL divergence KL(g0 || g1) with g0 and g1 Gamma.
Args:
g0: instance of a Gamma distribution object.
g1: instance of a Gamma distribution object.
name: (optional) Name to use for created operations.
Default is "kl_gamma_gamma".
Returns:
kl_gamma_gamma: `Tensor`. The batchwise KL(g0 || g1).
"""
with ops.name_scope(name, "kl_gamma_gamma", values=[
g0.concentration, g0.rate, g1.concentration, g1.rate]):
# Result from:
# http://www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps
# For derivation see:
# http://stats.stackexchange.com/questions/11646/kullback-leibler-divergence-between-two-gamma-distributions pylint: disable=line-too-long
return (((g0.concentration - g1.concentration)
* math_ops.digamma(g0.concentration))
+ math_ops.lgamma(g1.concentration)
- math_ops.lgamma(g0.concentration)
+ g1.concentration * math_ops.log(g0.rate)
- g1.concentration * math_ops.log(g1.rate)
+ g0.concentration * (g1.rate / g0.rate - 1.))
tensorflow/python/ops/distributions/identity_bijector.py
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# 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
# limitations under the License.
# ==============================================================================
"""Identity bijector."""
from tensorflow.python.framework import constant_op
from tensorflow.python.ops.distributions import bijector
from tensorflow.python.util import deprecation
__all__ = [
"Identity",
]
class Identity(bijector.Bijector):
"""Compute Y = g(X) = X.
Example Use:
```python
# Create the Y=g(X)=X transform which is intended for Tensors with 1 batch
# ndim and 1 event ndim (i.e., vector of vectors).
identity = Identity()
x = [[1., 2],
[3, 4]]
x == identity.forward(x) == identity.inverse(x)
```
"""
@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, validate_args=False, name="identity"):
super(Identity, self).__init__(
forward_min_event_ndims=0,
is_constant_jacobian=True,
validate_args=validate_args,
name=name)
def _forward(self, x):
return x
def _inverse(self, y):
return y
def _inverse_log_det_jacobian(self, y):
return constant_op.constant(0., dtype=y.dtype)
def _forward_log_det_jacobian(self, x):
return constant_op.constant(0., dtype=x.dtype)
tensorflow/python/ops/distributions/kullback_leibler.py
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# 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
# limitations under the License.
# ==============================================================================
"""Registration and usage mechanisms for KL-divergences."""
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import control_flow_assert
from tensorflow.python.ops import math_ops
from tensorflow.python.util import deprecation
from tensorflow.python.util import tf_inspect
from tensorflow.python.util.tf_export import tf_export
_DIVERGENCES = {}
__all__ = [
"RegisterKL",
"kl_divergence",
]
def _registered_kl(type_a, type_b):
"""Get the KL function registered for classes a and b."""
hierarchy_a = tf_inspect.getmro(type_a)
hierarchy_b = tf_inspect.getmro(type_b)
dist_to_children = None
kl_fn = None
for mro_to_a, parent_a in enumerate(hierarchy_a):
for mro_to_b, parent_b in enumerate(hierarchy_b):
candidate_dist = mro_to_a + mro_to_b
candidate_kl_fn = _DIVERGENCES.get((parent_a, parent_b), None)
if not kl_fn or (candidate_kl_fn and candidate_dist < dist_to_children):
dist_to_children = candidate_dist
kl_fn = candidate_kl_fn
return kl_fn
@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)
@tf_export(v1=["distributions.kl_divergence"])
def kl_divergence(distribution_a, distribution_b,
allow_nan_stats=True, name=None):
"""Get the KL-divergence KL(distribution_a || distribution_b).
If there is no KL method registered specifically for `type(distribution_a)`
and `type(distribution_b)`, then the class hierarchies of these types are
searched.
If one KL method is registered between any pairs of classes in these two
parent hierarchies, it is used.
If more than one such registered method exists, the method whose registered
classes have the shortest sum MRO paths to the input types is used.
If more than one such shortest path exists, the first method
identified in the search is used (favoring a shorter MRO distance to
`type(distribution_a)`).
Args:
distribution_a: The first distribution.
distribution_b: The second distribution.
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.
Returns:
A Tensor with the batchwise KL-divergence between `distribution_a`
and `distribution_b`.
Raises:
NotImplementedError: If no KL method is defined for distribution types
of `distribution_a` and `distribution_b`.
"""
kl_fn = _registered_kl(type(distribution_a), type(distribution_b))
if kl_fn is None:
raise NotImplementedError(
"No KL(distribution_a || distribution_b) registered for distribution_a "
"type %s and distribution_b type %s"
% (type(distribution_a).__name__, type(distribution_b).__name__))
with ops.name_scope("KullbackLeibler"):
kl_t = kl_fn(distribution_a, distribution_b, name=name)
if allow_nan_stats:
return kl_t
# Check KL for NaNs
kl_t = array_ops.identity(kl_t, name="kl")
with ops.control_dependencies([
control_flow_assert.Assert(
math_ops.logical_not(math_ops.reduce_any(math_ops.is_nan(kl_t))), [
"KL calculation between %s and %s returned NaN values "
"(and was called with allow_nan_stats=False). Values:" %
(distribution_a.name, distribution_b.name), kl_t
])
]):
return array_ops.identity(kl_t, name="checked_kl")
@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 cross_entropy(ref, other,
allow_nan_stats=True, name=None):
"""Computes the (Shannon) cross entropy.
Denote two distributions by `P` (`ref`) and `Q` (`other`). Assuming `P, Q`
are absolutely continuous with respect to one another and permit densities
`p(x) dr(x)` and `q(x) dr(x)`, (Shanon) cross entropy is defined as:
```none
H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
```
where `F` denotes the support of the random variable `X ~ P`.
Args:
ref: `tfd.Distribution` instance.
other: `tfd.Distribution` instance.
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` prepended to names of ops created by this function.
Returns:
cross_entropy: `ref.dtype` `Tensor` with shape `[B1, ..., Bn]`
representing `n` different calculations of (Shanon) cross entropy.
"""
with ops.name_scope(name, "cross_entropy"):
return ref.entropy() + kl_divergence(
ref, other, allow_nan_stats=allow_nan_stats)
@tf_export(v1=["distributions.RegisterKL"])
class RegisterKL:
"""Decorator to register a KL divergence implementation function.
Usage:
@distributions.RegisterKL(distributions.Normal, distributions.Normal)
def _kl_normal_mvn(norm_a, norm_b):
# Return KL(norm_a || norm_b)
"""
@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, dist_cls_a, dist_cls_b):
"""Initialize the KL registrar.
Args:
dist_cls_a: the class of the first argument of the KL divergence.
dist_cls_b: the class of the second argument of the KL divergence.
"""
self._key = (dist_cls_a, dist_cls_b)
def __call__(self, kl_fn):
"""Perform the KL registration.
Args:
kl_fn: The function to use for the KL divergence.
Returns:
kl_fn
Raises:
TypeError: if kl_fn is not a callable.
ValueError: if a KL divergence function has already been registered for
the given argument classes.
"""
if not callable(kl_fn):
raise TypeError("kl_fn must be callable, received: %s" % kl_fn)
if self._key in _DIVERGENCES:
raise ValueError("KL(%s || %s) has already been registered to: %s"
% (self._key[0].__name__, self._key[1].__name__,
_DIVERGENCES[self._key]))
_DIVERGENCES[self._key] = kl_fn
return kl_fn
tensorflow/python/ops/distributions/laplace.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Laplace distribution class."""
import math
import numpy as np
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import check_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import nn
from tensorflow.python.ops import random_ops
from tensorflow.python.ops.distributions import distribution
from tensorflow.python.ops.distributions import special_math
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export
__all__ = [
"Laplace",
"LaplaceWithSoftplusScale",
]
@tf_export(v1=["distributions.Laplace"])
class Laplace(distribution.Distribution):
"""The Laplace distribution with location `loc` and `scale` parameters.
#### Mathematical details
The probability density function (pdf) of this distribution is,
```none
pdf(x; mu, sigma) = exp(-|x - mu| / sigma) / Z
Z = 2 sigma
```
where `loc = mu`, `scale = sigma`, and `Z` is the normalization constant.
Note that the Laplace distribution can be thought of two exponential
distributions spliced together "back-to-back."
The Lpalce 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 ~ Laplace(loc=0, scale=1)
Y = loc + scale * X
```
"""
@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,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="Laplace"):
"""Construct Laplace distribution with parameters `loc` and `scale`.
The parameters `loc` and `scale` must be shaped in a way that supports
broadcasting (e.g., `loc / scale` is a valid operation).
Args:
loc: Floating point tensor which characterizes the location (center)
of the distribution.
scale: Positive floating point tensor which characterizes the spread of
the distribution.
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:
TypeError: if `loc` and `scale` are of different dtype.
"""
parameters = dict(locals())
with ops.name_scope(name, values=[loc, scale]) as name:
with ops.control_dependencies([check_ops.assert_positive(scale)] if
validate_args else []):
self._loc = array_ops.identity(loc, name="loc")
self._scale = array_ops.identity(scale, name="scale")
check_ops.assert_same_float_dtype([self._loc, self._scale])
super(Laplace, self).__init__(
dtype=self._loc.dtype,
reparameterization_type=distribution.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._loc, self._scale],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return dict(
zip(("loc", "scale"), ([ops.convert_to_tensor(
sample_shape, dtype=dtypes.int32)] * 2)))
@property
def loc(self):
"""Distribution parameter for the location."""
return self._loc
@property
def scale(self):
"""Distribution parameter for scale."""
return self._scale
def _batch_shape_tensor(self):
return array_ops.broadcast_dynamic_shape(
array_ops.shape(self.loc), array_ops.shape(self.scale))
def _batch_shape(self):
return array_ops.broadcast_static_shape(
self.loc.get_shape(), self.scale.get_shape())
def _event_shape_tensor(self):
return constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
# Uniform variates must be sampled from the open-interval `(-1, 1)` rather
# than `[-1, 1)`. In the case of `(0, 1)` we'd use
# `np.finfo(self.dtype.as_numpy_dtype).tiny` because it is the smallest,
# positive, "normal" number. However, the concept of subnormality exists
# only at zero; here we need the smallest usable number larger than -1,
# i.e., `-1 + eps/2`.
uniform_samples = random_ops.random_uniform(
shape=shape,
minval=np.nextafter(self.dtype.as_numpy_dtype(-1.),
self.dtype.as_numpy_dtype(0.)),
maxval=1.,
dtype=self.dtype,
seed=seed)
return (self.loc - self.scale * math_ops.sign(uniform_samples) *
math_ops.log1p(-math_ops.abs(uniform_samples)))
def _log_prob(self, x):
return self._log_unnormalized_prob(x) - self._log_normalization()
def _prob(self, x):
return math_ops.exp(self._log_prob(x))
def _log_cdf(self, x):
return special_math.log_cdf_laplace(self._z(x))
def _log_survival_function(self, x):
return special_math.log_cdf_laplace(-self._z(x))
def _cdf(self, x):
z = self._z(x)
return (0.5 + 0.5 * math_ops.sign(z) *
(1. - math_ops.exp(-math_ops.abs(z))))
def _log_unnormalized_prob(self, x):
return -math_ops.abs(self._z(x))
def _log_normalization(self):
return math.log(2.) + math_ops.log(self.scale)
def _entropy(self):
# Use broadcasting rules to calculate the full broadcast scale.
scale = self.scale + array_ops.zeros_like(self.loc)
return math.log(2.) + 1. + math_ops.log(scale)
def _mean(self):
return self.loc + array_ops.zeros_like(self.scale)
def _stddev(self):
return math.sqrt(2.) * self.scale + array_ops.zeros_like(self.loc)
def _median(self):
return self._mean()
def _mode(self):
return self._mean()
def _z(self, x):
return (x - self.loc) / self.scale
class LaplaceWithSoftplusScale(Laplace):
"""Laplace with softplus applied to `scale`."""
@deprecation.deprecated(
"2019-01-01",
"Use `tfd.Laplace(loc, tf.nn.softplus(scale)) "
"instead.",
warn_once=True)
def __init__(self,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="LaplaceWithSoftplusScale"):
parameters = dict(locals())
with ops.name_scope(name, values=[loc, scale]) as name:
super(LaplaceWithSoftplusScale, self).__init__(
loc=loc,
scale=nn.softplus(scale, name="softplus_scale"),
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
tensorflow/python/ops/distributions/multinomial.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Multinomial distribution class."""
from tensorflow.python.framework import dtypes
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 map_fn
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import nn_ops
from tensorflow.python.ops import random_ops
from tensorflow.python.ops.distributions import distribution
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__ = [
"Multinomial",
]
_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 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.probs` and `self.total_count`."""
@tf_export(v1=["distributions.Multinomial"])
class Multinomial(distribution.Distribution):
"""Multinomial distribution.
This Multinomial distribution is parameterized by `probs`, a (batch of)
length-`K` `prob` (probability) vectors (`K > 1`) such that
`tf.reduce_sum(probs, -1) = 1`, and a `total_count` number of trials, i.e.,
the number of trials per draw from the Multinomial. It is defined over a
(batch of) length-`K` vector `counts` such that
`tf.reduce_sum(counts, -1) = total_count`. The Multinomial is identically the
Binomial distribution when `K = 2`.
#### Mathematical Details
The 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; pi, N) = prod_j (pi_j)**n_j / Z
Z = (prod_j n_j!) / N!
```
where:
* `probs = pi = [pi_0, ..., pi_{K-1}]`, `pi_j > 0`, `sum_j pi_j = 1`,
* `total_count = N`, `N` a positive integer,
* `Z` is the normalization constant, and,
* `N!` denotes `N` factorial.
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
Create a 3-class distribution, with the 3rd class is most likely to be drawn,
using logits.
```python
logits = [-50., -43, 0]
dist = Multinomial(total_count=4., logits=logits)
```
Create a 3-class distribution, with the 3rd class is most likely to be drawn.
```python
p = [.2, .3, .5]
dist = Multinomial(total_count=4., probs=p)
```
The distribution functions can be evaluated on counts.
```python
# counts same shape as p.
counts = [1., 0, 3]
dist.prob(counts) # Shape []
# p will be broadcast to [[.2, .3, .5], [.2, .3, .5]] to match counts.
counts = [[1., 2, 1], [2, 2, 0]]
dist.prob(counts) # Shape [2]
# p will be broadcast to shape [5, 7, 3] to match counts.
counts = [[...]] # Shape [5, 7, 3]
dist.prob(counts) # Shape [5, 7]
```
Create a 2-batch of 3-class distributions.
```python
p = [[.1, .2, .7], [.3, .3, .4]] # Shape [2, 3]
dist = Multinomial(total_count=[4., 5], probs=p)
counts = [[2., 1, 1], [3, 1, 1]]
dist.prob(counts) # Shape [2]
dist.sample(5) # Shape [5, 2, 3]
```
"""
@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,
total_count,
logits=None,
probs=None,
validate_args=False,
allow_nan_stats=True,
name="Multinomial"):
"""Initialize a batch of Multinomial distributions.
Args:
total_count: Non-negative floating point tensor with shape broadcastable
to `[N1,..., Nm]` with `m >= 0`. Defines this as a batch of
`N1 x ... x Nm` different Multinomial distributions. Its components
should be equal to integer values.
logits: Floating point tensor representing unnormalized log-probabilities
of a positive event with shape broadcastable to
`[N1,..., Nm, K]` `m >= 0`, and the same dtype as `total_count`. Defines
this as a batch of `N1 x ... x Nm` different `K` class Multinomial
distributions. Only one of `logits` or `probs` should be passed in.
probs: Positive floating point tensor with shape broadcastable to
`[N1,..., Nm, K]` `m >= 0` and same dtype as `total_count`. Defines
this as a batch of `N1 x ... x Nm` different `K` class Multinomial
distributions. `probs`'s components in the last portion of its shape
should sum to `1`. Only one of `logits` or `probs` should be passed in.
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=[total_count, logits, probs]) as name:
self._total_count = ops.convert_to_tensor(total_count, name="total_count")
if validate_args:
self._total_count = (
distribution_util.embed_check_nonnegative_integer_form(
self._total_count))
self._logits, self._probs = distribution_util.get_logits_and_probs(
logits=logits,
probs=probs,
multidimensional=True,
validate_args=validate_args,
name=name)
self._mean_val = self._total_count[..., array_ops.newaxis] * self._probs
super(Multinomial, self).__init__(
dtype=self._probs.dtype,
reparameterization_type=distribution.NOT_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._total_count,
self._logits,
self._probs],
name=name)
@property
def total_count(self):
"""Number of trials used to construct a sample."""
return self._total_count
@property
def logits(self):
"""Vector of coordinatewise logits."""
return self._logits
@property
def probs(self):
"""Probability of drawing a `1` in that coordinate."""
return self._probs
def _batch_shape_tensor(self):
return array_ops.shape(self._mean_val)[:-1]
def _batch_shape(self):
return self._mean_val.get_shape().with_rank_at_least(1)[:-1]
def _event_shape_tensor(self):
return array_ops.shape(self._mean_val)[-1:]
def _event_shape(self):
return self._mean_val.get_shape().with_rank_at_least(1)[-1:]
def _sample_n(self, n, seed=None):
n_draws = math_ops.cast(self.total_count, dtype=dtypes.int32)
k = self.event_shape_tensor()[0]
# broadcast the total_count and logits to same shape
n_draws = array_ops.ones_like(
self.logits[..., 0], dtype=n_draws.dtype) * n_draws
logits = array_ops.ones_like(
n_draws[..., array_ops.newaxis], dtype=self.logits.dtype) * self.logits
# flatten the total_count and logits
flat_logits = array_ops.reshape(logits, [-1, k]) # [B1B2...Bm, k]
flat_ndraws = n * array_ops.reshape(n_draws, [-1]) # [B1B2...Bm]
# computes each total_count and logits situation by map_fn
def _sample_single(args):
logits, n_draw = args[0], args[1] # [K], []
x = random_ops.multinomial(logits[array_ops.newaxis, ...], n_draw,
seed) # [1, n*n_draw]
x = array_ops.reshape(x, shape=[n, -1]) # [n, n_draw]
x = math_ops.reduce_sum(array_ops.one_hot(x, depth=k), axis=-2) # [n, k]
return x
x = map_fn.map_fn(
_sample_single, [flat_logits, flat_ndraws],
dtype=self.dtype) # [B1B2...Bm, n, k]
# reshape the results to proper shape
x = array_ops.transpose(x, perm=[1, 0, 2])
final_shape = array_ops.concat([[n], self.batch_shape_tensor(), [k]], 0)
x = array_ops.reshape(x, final_shape) # [n, B1, B2,..., Bm, k]
return x
@distribution_util.AppendDocstring(_multinomial_sample_note)
def _log_prob(self, counts):
return self._log_unnormalized_prob(counts) - self._log_normalization(counts)
def _log_unnormalized_prob(self, counts):
counts = self._maybe_assert_valid_sample(counts)
return math_ops.reduce_sum(counts * nn_ops.log_softmax(self.logits), -1)
def _log_normalization(self, counts):
counts = self._maybe_assert_valid_sample(counts)
return -distribution_util.log_combinations(self.total_count, counts)
def _mean(self):
return array_ops.identity(self._mean_val)
def _covariance(self):
p = self.probs * array_ops.ones_like(
self.total_count)[..., array_ops.newaxis]
# pylint: disable=invalid-unary-operand-type
return array_ops.matrix_set_diag(
-math_ops.matmul(
self._mean_val[..., array_ops.newaxis],
p[..., array_ops.newaxis, :]), # outer product
self._variance())
def _variance(self):
p = self.probs * array_ops.ones_like(
self.total_count)[..., array_ops.newaxis]
return self._mean_val - self._mean_val * p
def _maybe_assert_valid_sample(self, counts):
"""Check counts for proper shape, values, then return tensor version."""
if not self.validate_args:
return counts
counts = distribution_util.embed_check_nonnegative_integer_form(counts)
return control_flow_ops.with_dependencies([
check_ops.assert_equal(
self.total_count, math_ops.reduce_sum(counts, -1),
message="counts must sum to `self.total_count`"),
], counts)
tensorflow/python/ops/distributions/normal.py
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# 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
# limitations under the License.
# ==============================================================================
"""The Normal (Gaussian) distribution class."""
import math
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import check_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import nn
from tensorflow.python.ops import random_ops
from tensorflow.python.ops.distributions import distribution
from tensorflow.python.ops.distributions import kullback_leibler
from tensorflow.python.ops.distributions import special_math
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export
__all__ = [
"Normal",
"NormalWithSoftplusScale",
]
@tf_export(v1=["distributions.Normal"])
class Normal(distribution.Distribution):
"""The Normal distribution with location `loc` and `scale` parameters.
#### Mathematical details
The probability density function (pdf) is,
```none
pdf(x; mu, sigma) = exp(-0.5 (x - mu)**2 / sigma**2) / Z
Z = (2 pi sigma**2)**0.5
```
where `loc = mu` is the mean, `scale = sigma` is the std. deviation, and, `Z`
is the normalization constant.
The Normal 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 ~ Normal(loc=0, scale=1)
Y = loc + scale * X
```
#### Examples
Examples of initialization of one or a batch of distributions.
```python
import tensorflow_probability as tfp
tfd = tfp.distributions
# Define a single scalar Normal distribution.
dist = tfd.Normal(loc=0., scale=3.)
# Evaluate the cdf at 1, returning a scalar.
dist.cdf(1.)
# Define a batch of two scalar valued Normals.
# The first has mean 1 and standard deviation 11, the second 2 and 22.
dist = tfd.Normal(loc=[1, 2.], scale=[11, 22.])
# Evaluate the pdf of the first distribution on 0, and the second on 1.5,
# returning a length two tensor.
dist.prob([0, 1.5])
# Get 3 samples, returning a 3 x 2 tensor.
dist.sample([3])
```
Arguments are broadcast when possible.
```python
# Define a batch of two scalar valued Normals.
# Both have mean 1, but different standard deviations.
dist = tfd.Normal(loc=1., scale=[11, 22.])
# Evaluate the pdf of both distributions on the same point, 3.0,
# returning a length 2 tensor.
dist.prob(3.0)
```
"""
@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,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="Normal"):
"""Construct Normal distributions with mean and stddev `loc` and `scale`.
The parameters `loc` and `scale` must be shaped in a way that supports
broadcasting (e.g. `loc + scale` is a valid operation).
Args:
loc: Floating point tensor; the means of the distribution(s).
scale: Floating point tensor; the stddevs of the distribution(s).
Must contain only positive values.
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:
TypeError: if `loc` and `scale` have different `dtype`.
"""
parameters = dict(locals())
with ops.name_scope(name, values=[loc, scale]) as name:
with ops.control_dependencies([check_ops.assert_positive(scale)] if
validate_args else []):
self._loc = array_ops.identity(loc, name="loc")
self._scale = array_ops.identity(scale, name="scale")
check_ops.assert_same_float_dtype([self._loc, self._scale])
super(Normal, self).__init__(
dtype=self._scale.dtype,
reparameterization_type=distribution.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._loc, self._scale],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return dict(
zip(("loc", "scale"), ([ops.convert_to_tensor(
sample_shape, dtype=dtypes.int32)] * 2)))
@property
def loc(self):
"""Distribution parameter for the mean."""
return self._loc
@property
def scale(self):
"""Distribution parameter for standard deviation."""
return self._scale
def _batch_shape_tensor(self):
return array_ops.broadcast_dynamic_shape(
array_ops.shape(self.loc),
array_ops.shape(self.scale))
def _batch_shape(self):
return array_ops.broadcast_static_shape(
self.loc.get_shape(),
self.scale.get_shape())
def _event_shape_tensor(self):
return constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
sampled = random_ops.random_normal(
shape=shape, mean=0., stddev=1., dtype=self.loc.dtype, seed=seed)
return sampled * self.scale + self.loc
def _log_prob(self, x):
return self._log_unnormalized_prob(x) - self._log_normalization()
def _log_cdf(self, x):
return special_math.log_ndtr(self._z(x))
def _cdf(self, x):
return special_math.ndtr(self._z(x))
def _log_survival_function(self, x):
return special_math.log_ndtr(-self._z(x))
def _survival_function(self, x):
return special_math.ndtr(-self._z(x))
def _log_unnormalized_prob(self, x):
return -0.5 * math_ops.square(self._z(x))
def _log_normalization(self):
return 0.5 * math.log(2. * math.pi) + math_ops.log(self.scale)
def _entropy(self):
# Use broadcasting rules to calculate the full broadcast scale.
scale = self.scale * array_ops.ones_like(self.loc)
return 0.5 * math.log(2. * math.pi * math.e) + math_ops.log(scale)
def _mean(self):
return self.loc * array_ops.ones_like(self.scale)
def _quantile(self, p):
return self._inv_z(special_math.ndtri(p))
def _stddev(self):
return self.scale * array_ops.ones_like(self.loc)
def _mode(self):
return self._mean()
def _z(self, x):
"""Standardize input `x` to a unit normal."""
with ops.name_scope("standardize", values=[x]):
return (x - self.loc) / self.scale
def _inv_z(self, z):
"""Reconstruct input `x` from a its normalized version."""
with ops.name_scope("reconstruct", values=[z]):
return z * self.scale + self.loc
class NormalWithSoftplusScale(Normal):
"""Normal with softplus applied to `scale`."""
@deprecation.deprecated(
"2019-01-01",
"Use `tfd.Normal(loc, tf.nn.softplus(scale)) "
"instead.",
warn_once=True)
def __init__(self,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="NormalWithSoftplusScale"):
parameters = dict(locals())
with ops.name_scope(name, values=[scale]) as name:
super(NormalWithSoftplusScale, self).__init__(
loc=loc,
scale=nn.softplus(scale, name="softplus_scale"),
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
@kullback_leibler.RegisterKL(Normal, Normal)
def _kl_normal_normal(n_a, n_b, name=None):
"""Calculate the batched KL divergence KL(n_a || n_b) with n_a and n_b Normal.
Args:
n_a: instance of a Normal distribution object.
n_b: instance of a Normal distribution object.
name: (optional) Name to use for created operations.
default is "kl_normal_normal".
Returns:
Batchwise KL(n_a || n_b)
"""
with ops.name_scope(name, "kl_normal_normal", [n_a.loc, n_b.loc]):
one = constant_op.constant(1, dtype=n_a.dtype)
two = constant_op.constant(2, dtype=n_a.dtype)
half = constant_op.constant(0.5, dtype=n_a.dtype)
s_a_squared = math_ops.square(n_a.scale)
s_b_squared = math_ops.square(n_b.scale)
ratio = s_a_squared / s_b_squared
return (math_ops.squared_difference(n_a.loc, n_b.loc) / (two * s_b_squared)
+ half * (ratio - one - math_ops.log(ratio)))
tensorflow/python/ops/distributions/special_math.py
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# 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
# limitations under the License.
# ==============================================================================
# Functions "ndtr" and "ndtri" are derived from calculations made in:
# https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
# In the following email exchange, the author gives his consent to redistribute
# derived works under an Apache 2.0 license.
#
# From: Stephen Moshier <steve@moshier.net>
# Date: Sat, Jun 9, 2018 at 2:36 PM
# Subject: Re: Licensing cephes under Apache (BSD-like) license.
# To: rif <rif@google.com>
#
#
#
# Hello Rif,
#
# Yes, Google may distribute Cephes files under the Apache 2 license.
#
# If clarification is needed, I do not favor BSD over other free licenses.
# I would agree that Apache 2 seems to cover the concern you mentioned
# about sublicensees.
#
# Best wishes for good luck with your projects!
# Steve Moshier
#
#
#
# On Thu, 31 May 2018, rif wrote:
#
# > Hello Steve.
# > My name is Rif. I work on machine learning software at Google.
# >
# > Your cephes software continues to be incredibly useful and widely used. I
# > was wondering whether it would be permissible for us to use the Cephes code
# > under the Apache 2.0 license, which is extremely similar in permissions to
# > the BSD license (Wikipedia comparisons). This would be quite helpful to us
# > in terms of avoiding multiple licenses on software.
# >
# > I'm sorry to bother you with this (I can imagine you're sick of hearing
# > about this by now), but I want to be absolutely clear we're on the level and
# > not misusing your important software. In former conversation with Eugene
# > Brevdo (ebrevdo@google.com), you wrote "If your licensing is similar to BSD,
# > the formal way that has been handled is simply to add a statement to the
# > effect that you are incorporating the Cephes software by permission of the
# > author." I wanted to confirm that (a) we could use the Apache license, (b)
# > that we don't need to (and probably you don't want to) keep getting
# > contacted about individual uses, because your intent is generally to allow
# > this software to be reused under "BSD-like" license, and (c) you're OK
# > letting incorporators decide whether a license is sufficiently BSD-like?
# >
# > Best,
# >
# > rif
# >
# >
# >
"""Special Math Ops."""
import numpy as np
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import math_ops
__all__ = [
"erfinv",
"ndtr",
"ndtri",
"log_ndtr",
"log_cdf_laplace",
]
# log_ndtr uses different functions over the ranges
# (-infty, lower](lower, upper](upper, infty)
# Lower bound values were chosen by examining where the support of ndtr
# appears to be zero, relative to scipy's (which is always 64bit). They were
# then made more conservative just to be safe. (Conservative means use the
# expansion more than we probably need to.) See `NdtrTest` in
# special_math_test.py.
LOGNDTR_FLOAT64_LOWER = np.array(-20, np.float64)
LOGNDTR_FLOAT32_LOWER = np.array(-10, np.float32)
# Upper bound values were chosen by examining for which values of 'x'
# Log[cdf(x)] is 0, after which point we need to use the approximation
# Log[cdf(x)] = Log[1 - cdf(-x)] approx -cdf(-x). We chose a value slightly
# conservative, meaning we use the approximation earlier than needed.
LOGNDTR_FLOAT64_UPPER = np.array(8, np.float64)
LOGNDTR_FLOAT32_UPPER = np.array(5, np.float32)
def ndtr(x, name="ndtr"):
"""Normal distribution function.
Returns the area under the Gaussian probability density function, integrated
from minus infinity to x:
```
1 / x
ndtr(x) = ---------- | exp(-0.5 t**2) dt
sqrt(2 pi) /-inf
= 0.5 (1 + erf(x / sqrt(2)))
= 0.5 erfc(x / sqrt(2))
```
Args:
x: `Tensor` of type `float32`, `float64`.
name: Python string. A name for the operation (default="ndtr").
Returns:
ndtr: `Tensor` with `dtype=x.dtype`.
Raises:
TypeError: if `x` is not floating-type.
"""
with ops.name_scope(name, values=[x]):
x = ops.convert_to_tensor(x, name="x")
if x.dtype.as_numpy_dtype not in [np.float32, np.float64]:
raise TypeError(
"x.dtype=%s is not handled, see docstring for supported types."
% x.dtype)
return _ndtr(x)
def _ndtr(x):
"""Implements ndtr core logic."""
half_sqrt_2 = constant_op.constant(
0.5 * np.sqrt(2.), dtype=x.dtype, name="half_sqrt_2")
w = x * half_sqrt_2
z = math_ops.abs(w)
y = array_ops.where_v2(
math_ops.less(z, half_sqrt_2), 1. + math_ops.erf(w),
array_ops.where_v2(
math_ops.greater(w, 0.), 2. - math_ops.erfc(z), math_ops.erfc(z)))
return 0.5 * y
def ndtri(p, name="ndtri"):
"""The inverse of the CDF of the Normal distribution function.
Returns x such that the area under the pdf from minus infinity to x is equal
to p.
A piece-wise rational approximation is done for the function.
This is a port of the implementation in netlib.
Args:
p: `Tensor` of type `float32`, `float64`.
name: Python string. A name for the operation (default="ndtri").
Returns:
x: `Tensor` with `dtype=p.dtype`.
Raises:
TypeError: if `p` is not floating-type.
"""
with ops.name_scope(name, values=[p]):
p = ops.convert_to_tensor(p, name="p")
if p.dtype.as_numpy_dtype not in [np.float32, np.float64]:
raise TypeError(
"p.dtype=%s is not handled, see docstring for supported types."
% p.dtype)
return _ndtri(p)
def _ndtri(p):
"""Implements ndtri core logic."""
# Constants used in piece-wise rational approximations. Taken from the cephes
# library:
# https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
p0 = [
-1.23916583867381258016E0, 1.39312609387279679503E1,
-5.66762857469070293439E1, 9.80010754185999661536E1,
-5.99633501014107895267E1
]
q0 = [
-1.18331621121330003142E0, 1.59056225126211695515E1,
-8.20372256168333339912E1, 2.00260212380060660359E2,
-2.25462687854119370527E2, 8.63602421390890590575E1,
4.67627912898881538453E0, 1.95448858338141759834E0, 1.0
]
p1 = [
-8.57456785154685413611E-4, -3.50424626827848203418E-2,
-1.40256079171354495875E-1, 2.18663306850790267539E0,
1.46849561928858024014E1, 4.40805073893200834700E1,
5.71628192246421288162E1, 3.15251094599893866154E1,
4.05544892305962419923E0
]
q1 = [
-9.33259480895457427372E-4, -3.80806407691578277194E-2,
-1.42182922854787788574E-1, 2.50464946208309415979E0,
1.50425385692907503408E1, 4.13172038254672030440E1,
4.53907635128879210584E1, 1.57799883256466749731E1, 1.0
]
p2 = [
6.23974539184983293730E-9, 2.65806974686737550832E-6,
3.01581553508235416007E-4, 1.23716634817820021358E-2,
2.01485389549179081538E-1, 1.33303460815807542389E0,
3.93881025292474443415E0, 6.91522889068984211695E0,
3.23774891776946035970E0
]
q2 = [
6.79019408009981274425E-9, 2.89247864745380683936E-6,
3.28014464682127739104E-4, 1.34204006088543189037E-2,
2.16236993594496635890E-1, 1.37702099489081330271E0,
3.67983563856160859403E0, 6.02427039364742014255E0, 1.0
]
def _create_polynomial(var, coeffs):
"""Compute n_th order polynomial via Horner's method."""
coeffs = np.array(coeffs, var.dtype.as_numpy_dtype)
if not coeffs.size:
return array_ops.zeros_like(var)
return coeffs[0] + _create_polynomial(var, coeffs[1:]) * var
maybe_complement_p = array_ops.where_v2(p > -np.expm1(-2.), 1. - p, p)
# Write in an arbitrary value in place of 0 for p since 0 will cause NaNs
# later on. The result from the computation when p == 0 is not used so any
# number that doesn't result in NaNs is fine.
sanitized_mcp = array_ops.where_v2(
maybe_complement_p <= 0.,
array_ops.fill(array_ops.shape(p), np.array(0.5, p.dtype.as_numpy_dtype)),
maybe_complement_p)
# Compute x for p > exp(-2): x/sqrt(2pi) = w + w**3 P0(w**2)/Q0(w**2).
w = sanitized_mcp - 0.5
ww = w ** 2
x_for_big_p = w + w * ww * (_create_polynomial(ww, p0)
/ _create_polynomial(ww, q0))
x_for_big_p *= -np.sqrt(2. * np.pi)
# Compute x for p <= exp(-2): x = z - log(z)/z - (1/z) P(1/z) / Q(1/z),
# where z = sqrt(-2. * log(p)), and P/Q are chosen between two different
# arrays based on whether p < exp(-32).
z = math_ops.sqrt(-2. * math_ops.log(sanitized_mcp))
first_term = z - math_ops.log(z) / z
second_term_small_p = (
_create_polynomial(1. / z, p2) /
_create_polynomial(1. / z, q2) / z)
second_term_otherwise = (
_create_polynomial(1. / z, p1) /
_create_polynomial(1. / z, q1) / z)
x_for_small_p = first_term - second_term_small_p
x_otherwise = first_term - second_term_otherwise
x = array_ops.where_v2(
sanitized_mcp > np.exp(-2.), x_for_big_p,
array_ops.where_v2(z >= 8.0, x_for_small_p, x_otherwise))
x = array_ops.where_v2(p > 1. - np.exp(-2.), x, -x)
infinity_scalar = constant_op.constant(np.inf, dtype=p.dtype)
infinity = array_ops.fill(array_ops.shape(p), infinity_scalar)
x_nan_replaced = array_ops.where_v2(p <= 0.0, -infinity,
array_ops.where_v2(p >= 1.0, infinity, x))
return x_nan_replaced
def log_ndtr(x, series_order=3, name="log_ndtr"):
"""Log Normal distribution function.
For details of the Normal distribution function see `ndtr`.
This function calculates `(log o ndtr)(x)` by either calling `log(ndtr(x))` or
using an asymptotic series. Specifically:
- For `x > upper_segment`, use the approximation `-ndtr(-x)` based on
`log(1-x) ~= -x, x << 1`.
- For `lower_segment < x <= upper_segment`, use the existing `ndtr` technique
and take a log.
- For `x <= lower_segment`, we use the series approximation of erf to compute
the log CDF directly.
The `lower_segment` is set based on the precision of the input:
```
lower_segment = { -20, x.dtype=float64
{ -10, x.dtype=float32
upper_segment = { 8, x.dtype=float64
{ 5, x.dtype=float32
```
When `x < lower_segment`, the `ndtr` asymptotic series approximation is:
```
ndtr(x) = scale * (1 + sum) + R_N
scale = exp(-0.5 x**2) / (-x sqrt(2 pi))
sum = Sum{(-1)^n (2n-1)!! / (x**2)^n, n=1:N}
R_N = O(exp(-0.5 x**2) (2N+1)!! / |x|^{2N+3})
```
where `(2n-1)!! = (2n-1) (2n-3) (2n-5) ... (3) (1)` is a
[double-factorial](https://en.wikipedia.org/wiki/Double_factorial).
Args:
x: `Tensor` of type `float32`, `float64`.
series_order: Positive Python `integer`. Maximum depth to
evaluate the asymptotic expansion. This is the `N` above.
name: Python string. A name for the operation (default="log_ndtr").
Returns:
log_ndtr: `Tensor` with `dtype=x.dtype`.
Raises:
TypeError: if `x.dtype` is not handled.
TypeError: if `series_order` is a not Python `integer.`
ValueError: if `series_order` is not in `[0, 30]`.
"""
if not isinstance(series_order, int):
raise TypeError("series_order must be a Python integer.")
if series_order < 0:
raise ValueError("series_order must be non-negative.")
if series_order > 30:
raise ValueError("series_order must be <= 30.")
with ops.name_scope(name, values=[x]):
x = ops.convert_to_tensor(x, name="x")
if x.dtype.as_numpy_dtype == np.float64:
lower_segment = LOGNDTR_FLOAT64_LOWER
upper_segment = LOGNDTR_FLOAT64_UPPER
elif x.dtype.as_numpy_dtype == np.float32:
lower_segment = LOGNDTR_FLOAT32_LOWER
upper_segment = LOGNDTR_FLOAT32_UPPER
else:
raise TypeError("x.dtype=%s is not supported." % x.dtype)
# The basic idea here was ported from:
# https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
# We copy the main idea, with a few changes
# * For x >> 1, and X ~ Normal(0, 1),
# Log[P[X < x]] = Log[1 - P[X < -x]] approx -P[X < -x],
# which extends the range of validity of this function.
# * We use one fixed series_order for all of 'x', rather than adaptive.
# * Our docstring properly reflects that this is an asymptotic series, not a
# Taylor series. We also provided a correct bound on the remainder.
# * We need to use the max/min in the _log_ndtr_lower arg to avoid nan when
# x=0. This happens even though the branch is unchosen because when x=0
# the gradient of a select involves the calculation 1*dy+0*(-inf)=nan
# regardless of whether dy is finite. Note that the minimum is a NOP if
# the branch is chosen.
return array_ops.where_v2(
math_ops.greater(x, upper_segment),
-_ndtr(-x), # log(1-x) ~= -x, x << 1 # pylint: disable=invalid-unary-operand-type
array_ops.where_v2(
math_ops.greater(x, lower_segment),
math_ops.log(_ndtr(math_ops.maximum(x, lower_segment))),
_log_ndtr_lower(math_ops.minimum(x, lower_segment), series_order)))
def _log_ndtr_lower(x, series_order):
"""Asymptotic expansion version of `Log[cdf(x)]`, appropriate for `x<<-1`."""
x_2 = math_ops.square(x)
# Log of the term multiplying (1 + sum)
log_scale = -0.5 * x_2 - math_ops.log(-x) - 0.5 * np.log(2. * np.pi)
return log_scale + math_ops.log(_log_ndtr_asymptotic_series(x, series_order))
def _log_ndtr_asymptotic_series(x, series_order):
"""Calculates the asymptotic series used in log_ndtr."""
dtype = x.dtype.as_numpy_dtype
if series_order <= 0:
return np.array(1, dtype)
x_2 = math_ops.square(x)
even_sum = array_ops.zeros_like(x)
odd_sum = array_ops.zeros_like(x)
x_2n = x_2 # Start with x^{2*1} = x^{2*n} with n = 1.
for n in range(1, series_order + 1):
y = np.array(_double_factorial(2 * n - 1), dtype) / x_2n
if n % 2:
odd_sum += y
else:
even_sum += y
x_2n *= x_2
return 1. + even_sum - odd_sum
def erfinv(x, name="erfinv"):
"""The inverse function for erf, the error function.
Args:
x: `Tensor` of type `float32`, `float64`.
name: Python string. A name for the operation (default="erfinv").
Returns:
x: `Tensor` with `dtype=x.dtype`.
Raises:
TypeError: if `x` is not floating-type.
"""
with ops.name_scope(name, values=[x]):
x = ops.convert_to_tensor(x, name="x")
if x.dtype.as_numpy_dtype not in [np.float32, np.float64]:
raise TypeError(
"x.dtype=%s is not handled, see docstring for supported types."
% x.dtype)
return ndtri((x + 1.0) / 2.0) / np.sqrt(2)
def _double_factorial(n):
"""The double factorial function for small Python integer `n`."""
return np.prod(np.arange(n, 1, -2))
def log_cdf_laplace(x, name="log_cdf_laplace"):
"""Log Laplace distribution function.
This function calculates `Log[L(x)]`, where `L(x)` is the cumulative
distribution function of the Laplace distribution, i.e.
```L(x) := 0.5 * int_{-infty}^x e^{-|t|} dt```
For numerical accuracy, `L(x)` is computed in different ways depending on `x`,
```
x <= 0:
Log[L(x)] = Log[0.5] + x, which is exact
0 < x:
Log[L(x)] = Log[1 - 0.5 * e^{-x}], which is exact
```
Args:
x: `Tensor` of type `float32`, `float64`.
name: Python string. A name for the operation (default="log_ndtr").
Returns:
`Tensor` with `dtype=x.dtype`.
Raises:
TypeError: if `x.dtype` is not handled.
"""
with ops.name_scope(name, values=[x]):
x = ops.convert_to_tensor(x, name="x")
# For x < 0, L(x) = 0.5 * exp{x} exactly, so Log[L(x)] = log(0.5) + x.
lower_solution = -np.log(2.) + x
# safe_exp_neg_x = exp{-x} for x > 0, but is
# bounded above by 1, which avoids
# log[1 - 1] = -inf for x = log(1/2), AND
# exp{-x} --> inf, for x << -1
safe_exp_neg_x = math_ops.exp(-math_ops.abs(x))
# log1p(z) = log(1 + z) approx z for |z| << 1. This approximation is used
# internally by log1p, rather than being done explicitly here.
upper_solution = math_ops.log1p(-0.5 * safe_exp_neg_x)
return array_ops.where_v2(x < 0., lower_solution, upper_solution)
tensorflow/python/ops/distributions/student_t.py
+391
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# 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
# limitations under the License.
# ==============================================================================
"""Student's t distribution class."""
import numpy as np
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
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 nn
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 util as distribution_util
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export
__all__ = [
"StudentT",
"StudentTWithAbsDfSoftplusScale",
]
@tf_export(v1=["distributions.StudentT"])
class StudentT(distribution.Distribution):
"""Student's t-distribution.
This distribution has parameters: degree of freedom `df`, location `loc`,
and `scale`.
#### Mathematical details
The probability density function (pdf) is,
```none
pdf(x; df, mu, sigma) = (1 + y**2 / df)**(-0.5 (df + 1)) / Z
where,
y = (x - mu) / sigma
Z = abs(sigma) sqrt(df pi) Gamma(0.5 df) / Gamma(0.5 (df + 1))
```
where:
* `loc = mu`,
* `scale = sigma`, and,
* `Z` is the normalization constant, and,
* `Gamma` is the [gamma function](
https://en.wikipedia.org/wiki/Gamma_function).
The StudentT 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 ~ StudentT(df, loc=0, scale=1)
Y = loc + scale * X
```
Notice that `scale` has semantics more similar to standard deviation than
variance. However it is not actually the std. deviation; the Student's
t-distribution std. dev. is `scale sqrt(df / (df - 2))` when `df > 2`.
Samples of this distribution are reparameterized (pathwise differentiable).
The derivatives are computed using the approach described in
(Figurnov et al., 2018).
#### Examples
Examples of initialization of one or a batch of distributions.
```python
import tensorflow_probability as tfp
tfd = tfp.distributions
# Define a single scalar Student t distribution.
single_dist = tfd.StudentT(df=3)
# Evaluate the pdf at 1, returning a scalar Tensor.
single_dist.prob(1.)
# Define a batch of two scalar valued Student t's.
# The first has degrees of freedom 2, mean 1, and scale 11.
# The second 3, 2 and 22.
multi_dist = tfd.StudentT(df=[2, 3], loc=[1, 2.], scale=[11, 22.])
# Evaluate the pdf of the first distribution on 0, and the second on 1.5,
# returning a length two tensor.
multi_dist.prob([0, 1.5])
# Get 3 samples, returning a 3 x 2 tensor.
multi_dist.sample(3)
```
Arguments are broadcast when possible.
```python
# Define a batch of two Student's t distributions.
# Both have df 2 and mean 1, but different scales.
dist = tfd.StudentT(df=2, loc=1, scale=[11, 22.])
# Evaluate the pdf of both distributions on the same point, 3.0,
# returning a length 2 tensor.
dist.prob(3.0)
```
Compute the gradients of samples w.r.t. the parameters:
```python
df = tf.constant(2.0)
loc = tf.constant(2.0)
scale = tf.constant(11.0)
dist = tfd.StudentT(df=df, loc=loc, scale=scale)
samples = dist.sample(5) # Shape [5]
loss = tf.reduce_mean(tf.square(samples)) # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, [df, loc, scale])
```
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,
df,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="StudentT"):
"""Construct Student's t distributions.
The distributions have degree of freedom `df`, mean `loc`, and scale
`scale`.
The parameters `df`, `loc`, and `scale` must be shaped in a way that
supports broadcasting (e.g. `df + loc + scale` is a valid operation).
Args:
df: Floating-point `Tensor`. The degrees of freedom of the
distribution(s). `df` must contain only positive values.
loc: Floating-point `Tensor`. The mean(s) of the distribution(s).
scale: Floating-point `Tensor`. The scaling factor(s) for the
distribution(s). Note that `scale` is not technically the standard
deviation of this distribution but has semantics more similar to
standard deviation than variance.
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:
TypeError: if loc and scale are different dtypes.
"""
parameters = dict(locals())
with ops.name_scope(name, values=[df, loc, scale]) as name:
with ops.control_dependencies([check_ops.assert_positive(df)]
if validate_args else []):
self._df = array_ops.identity(df, name="df")
self._loc = array_ops.identity(loc, name="loc")
self._scale = array_ops.identity(scale, name="scale")
check_ops.assert_same_float_dtype(
(self._df, self._loc, self._scale))
super(StudentT, self).__init__(
dtype=self._scale.dtype,
reparameterization_type=distribution.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._df, self._loc, self._scale],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return dict(
zip(("df", "loc", "scale"), (
[ops.convert_to_tensor(
sample_shape, dtype=dtypes.int32)] * 3)))
@property
def df(self):
"""Degrees of freedom in these Student's t distribution(s)."""
return self._df
@property
def loc(self):
"""Locations of these Student's t distribution(s)."""
return self._loc
@property
def scale(self):
"""Scaling factors of these Student's t distribution(s)."""
return self._scale
def _batch_shape_tensor(self):
return array_ops.broadcast_dynamic_shape(
array_ops.shape(self.df),
array_ops.broadcast_dynamic_shape(
array_ops.shape(self.loc), array_ops.shape(self.scale)))
def _batch_shape(self):
return array_ops.broadcast_static_shape(
array_ops.broadcast_static_shape(self.df.get_shape(),
self.loc.get_shape()),
self.scale.get_shape())
def _event_shape_tensor(self):
return constant_op.constant([], dtype=math_ops.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
# The sampling method comes from the fact that if:
# X ~ Normal(0, 1)
# Z ~ Chi2(df)
# Y = X / sqrt(Z / df)
# then:
# Y ~ StudentT(df).
shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
normal_sample = random_ops.random_normal(shape, dtype=self.dtype, seed=seed)
df = self.df * array_ops.ones(self.batch_shape_tensor(), dtype=self.dtype)
gamma_sample = random_ops.random_gamma(
[n],
0.5 * df,
beta=0.5,
dtype=self.dtype,
seed=distribution_util.gen_new_seed(seed, salt="student_t"))
samples = normal_sample * math_ops.rsqrt(gamma_sample / df)
return samples * self.scale + self.loc # Abs(scale) not wanted.
def _log_prob(self, x):
return self._log_unnormalized_prob(x) - self._log_normalization()
def _log_unnormalized_prob(self, x):
y = (x - self.loc) / self.scale # Abs(scale) superfluous.
return -0.5 * (self.df + 1.) * math_ops.log1p(y**2. / self.df)
def _log_normalization(self):
return (math_ops.log(math_ops.abs(self.scale)) +
0.5 * math_ops.log(self.df) +
0.5 * np.log(np.pi) +
math_ops.lgamma(0.5 * self.df) -
math_ops.lgamma(0.5 * (self.df + 1.)))
def _cdf(self, x):
# Take Abs(scale) to make subsequent where work correctly.
y = (x - self.loc) / math_ops.abs(self.scale)
x_t = self.df / (y**2. + self.df)
neg_cdf = 0.5 * math_ops.betainc(0.5 * self.df, 0.5, x_t)
return array_ops.where_v2(math_ops.less(y, 0.), neg_cdf, 1. - neg_cdf)
def _entropy(self):
v = array_ops.ones(self.batch_shape_tensor(),
dtype=self.dtype)[..., array_ops.newaxis]
u = v * self.df[..., array_ops.newaxis]
beta_arg = array_ops.concat([u, v], -1) / 2.
return (math_ops.log(math_ops.abs(self.scale)) +
0.5 * math_ops.log(self.df) +
special_math_ops.lbeta(beta_arg) +
0.5 * (self.df + 1.) *
(math_ops.digamma(0.5 * (self.df + 1.)) -
math_ops.digamma(0.5 * self.df)))
@distribution_util.AppendDocstring(
"""The mean of Student's T equals `loc` if `df > 1`, otherwise it is
`NaN`. If `self.allow_nan_stats=True`, then an exception will be raised
rather than returning `NaN`.""")
def _mean(self):
mean = self.loc * array_ops.ones(self.batch_shape_tensor(),
dtype=self.dtype)
if self.allow_nan_stats:
nan = np.array(np.nan, dtype=self.dtype.as_numpy_dtype())
return array_ops.where_v2(
math_ops.greater(
self.df,
array_ops.ones(self.batch_shape_tensor(), dtype=self.dtype)),
mean, array_ops.fill(self.batch_shape_tensor(), nan, name="nan"))
else:
return control_flow_ops.with_dependencies(
[
check_ops.assert_less(
array_ops.ones([], dtype=self.dtype),
self.df,
message="mean not defined for components of df <= 1"),
],
mean)
@distribution_util.AppendDocstring("""
The variance for Student's T equals
```
df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1
```
""")
def _variance(self):
# We need to put the tf.where inside the outer tf.where to ensure we never
# hit a NaN in the gradient.
denom = array_ops.where_v2(
math_ops.greater(self.df, 2.), self.df - 2.,
array_ops.ones_like(self.df))
# Abs(scale) superfluous.
var = (array_ops.ones(self.batch_shape_tensor(), dtype=self.dtype) *
math_ops.square(self.scale) * self.df / denom)
# When 1 < df <= 2, variance is infinite.
inf = np.array(np.inf, dtype=self.dtype.as_numpy_dtype())
result_where_defined = array_ops.where_v2(
self.df > array_ops.fill(self.batch_shape_tensor(), 2.), var,
array_ops.fill(self.batch_shape_tensor(), inf, name="inf"))
if self.allow_nan_stats:
nan = np.array(np.nan, dtype=self.dtype.as_numpy_dtype())
return array_ops.where_v2(
math_ops.greater(
self.df,
array_ops.ones(self.batch_shape_tensor(), dtype=self.dtype)),
result_where_defined,
array_ops.fill(self.batch_shape_tensor(), nan, name="nan"))
else:
return control_flow_ops.with_dependencies(
[
check_ops.assert_less(
array_ops.ones([], dtype=self.dtype),
self.df,
message="variance not defined for components of df <= 1"),
],
result_where_defined)
def _mode(self):
return array_ops.identity(self.loc)
class StudentTWithAbsDfSoftplusScale(StudentT):
"""StudentT with `df = floor(abs(df))` and `scale = softplus(scale)`."""
@deprecation.deprecated(
"2019-01-01",
"Use `tfd.StudentT(tf.floor(tf.abs(df)), loc, "
"tf.nn.softplus(scale)) instead.",
warn_once=True)
def __init__(self,
df,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="StudentTWithAbsDfSoftplusScale"):
parameters = dict(locals())
with ops.name_scope(name, values=[df, scale]) as name:
super(StudentTWithAbsDfSoftplusScale, self).__init__(
df=math_ops.floor(math_ops.abs(df)),
loc=loc,
scale=nn.softplus(scale, name="softplus_scale"),
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
tensorflow/python/ops/distributions/transformed_distribution.py
+642
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# 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
# limitations under the License.
# ==============================================================================
"""A Transformed Distribution class."""
import numpy as np
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_util
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import array_ops_stack
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.distributions import distribution as distribution_lib
from tensorflow.python.ops.distributions import identity_bijector
from tensorflow.python.ops.distributions import util as distribution_util
from tensorflow.python.util import deprecation
__all__ = [
"TransformedDistribution",
]
# The following helper functions attempt to statically perform a TF operation.
# These functions make debugging easier since we can do more validation during
# graph construction.
def _static_value(x):
"""Returns the static value of a `Tensor` or `None`."""
return tensor_util.constant_value(ops.convert_to_tensor(x))
def _logical_and(*args):
"""Convenience function which attempts to statically `reduce_all`."""
args_ = [_static_value(x) for x in args]
if any(x is not None and not bool(x) for x in args_):
return constant_op.constant(False)
if all(x is not None and bool(x) for x in args_):
return constant_op.constant(True)
if len(args) == 2:
return math_ops.logical_and(*args)
return math_ops.reduce_all(args)
def _logical_equal(x, y):
"""Convenience function which attempts to statically compute `x == y`."""
x_ = _static_value(x)
y_ = _static_value(y)
if x_ is None or y_ is None:
return math_ops.equal(x, y)
return constant_op.constant(np.array_equal(x_, y_))
def _logical_not(x):
"""Convenience function which attempts to statically apply `logical_not`."""
x_ = _static_value(x)
if x_ is None:
return math_ops.logical_not(x)
return constant_op.constant(np.logical_not(x_))
def _concat_vectors(*args):
"""Convenience function which concatenates input vectors."""
args_ = [_static_value(x) for x in args]
if any(x_ is None for x_ in args_):
return array_ops.concat(args, 0)
return constant_op.constant([x_ for vec_ in args_ for x_ in vec_])
def _pick_scalar_condition(pred, cond_true, cond_false):
"""Convenience function which chooses the condition based on the predicate."""
# Note: This function is only valid if all of pred, cond_true, and cond_false
# are scalars. This means its semantics are arguably more like tf.cond than
# tf.select even though we use tf.select to implement it.
pred_ = _static_value(pred)
if pred_ is None:
return array_ops.where_v2(pred, cond_true, cond_false)
return cond_true if pred_ else cond_false
def _ones_like(x):
"""Convenience function attempts to statically construct `ones_like`."""
# Should only be used for small vectors.
if x.get_shape().is_fully_defined():
return array_ops.ones(x.get_shape().as_list(), dtype=x.dtype)
return array_ops.ones_like(x)
def _ndims_from_shape(shape):
"""Returns `Tensor`'s `rank` implied by a `Tensor` shape."""
if shape.get_shape().ndims not in (None, 1):
raise ValueError("input is not a valid shape: not 1D")
if not shape.dtype.is_integer:
raise TypeError("input is not a valid shape: wrong dtype")
if shape.get_shape().is_fully_defined():
return constant_op.constant(shape.get_shape().as_list()[0])
return array_ops.shape(shape)[0]
def _is_scalar_from_shape(shape):
"""Returns `True` `Tensor` if `Tensor` shape implies a scalar."""
return _logical_equal(_ndims_from_shape(shape), 0)
class TransformedDistribution(distribution_lib.Distribution):
"""A Transformed Distribution.
A `TransformedDistribution` models `p(y)` given a base distribution `p(x)`,
and a deterministic, invertible, differentiable transform, `Y = g(X)`. The
transform is typically an instance of the `Bijector` class and the base
distribution is typically an instance of the `Distribution` class.
A `Bijector` is expected to implement the following functions:
- `forward`,
- `inverse`,
- `inverse_log_det_jacobian`.
The semantics of these functions are outlined in the `Bijector` documentation.
We now describe how a `TransformedDistribution` alters the input/outputs of a
`Distribution` associated with a random variable (rv) `X`.
Write `cdf(Y=y)` for an absolutely continuous cumulative distribution function
of random variable `Y`; write the probability density function `pdf(Y=y) :=
d^k / (dy_1,...,dy_k) cdf(Y=y)` for its derivative wrt to `Y` evaluated at
`y`. Assume that `Y = g(X)` where `g` is a deterministic diffeomorphism,
i.e., a non-random, continuous, differentiable, and invertible function.
Write the inverse of `g` as `X = g^{-1}(Y)` and `(J o g)(x)` for the Jacobian
of `g` evaluated at `x`.
A `TransformedDistribution` implements the following operations:
* `sample`
Mathematically: `Y = g(X)`
Programmatically: `bijector.forward(distribution.sample(...))`
* `log_prob`
Mathematically: `(log o pdf)(Y=y) = (log o pdf o g^{-1})(y)
+ (log o abs o det o J o g^{-1})(y)`
Programmatically: `(distribution.log_prob(bijector.inverse(y))
+ bijector.inverse_log_det_jacobian(y))`
* `log_cdf`
Mathematically: `(log o cdf)(Y=y) = (log o cdf o g^{-1})(y)`
Programmatically: `distribution.log_cdf(bijector.inverse(x))`
* and similarly for: `cdf`, `prob`, `log_survival_function`,
`survival_function`.
A simple example constructing a Log-Normal distribution from a Normal
distribution:
```python
ds = tfp.distributions
log_normal = ds.TransformedDistribution(
distribution=ds.Normal(loc=0., scale=1.),
bijector=ds.bijectors.Exp(),
name="LogNormalTransformedDistribution")
```
A `LogNormal` made from callables:
```python
ds = tfp.distributions
log_normal = ds.TransformedDistribution(
distribution=ds.Normal(loc=0., scale=1.),
bijector=ds.bijectors.Inline(
forward_fn=tf.exp,
inverse_fn=tf.math.log,
inverse_log_det_jacobian_fn=(
lambda y: -tf.reduce_sum(tf.math.log(y), axis=-1)),
name="LogNormalTransformedDistribution")
```
Another example constructing a Normal from a StandardNormal:
```python
ds = tfp.distributions
normal = ds.TransformedDistribution(
distribution=ds.Normal(loc=0., scale=1.),
bijector=ds.bijectors.Affine(
shift=-1.,
scale_identity_multiplier=2.)
name="NormalTransformedDistribution")
```
A `TransformedDistribution`'s batch- and event-shape are implied by the base
distribution unless explicitly overridden by `batch_shape` or `event_shape`
arguments. Specifying an overriding `batch_shape` (`event_shape`) is
permitted only if the base distribution has scalar batch-shape (event-shape).
The bijector is applied to the distribution as if the distribution possessed
the overridden shape(s). The following example demonstrates how to construct a
multivariate Normal as a `TransformedDistribution`.
```python
ds = tfp.distributions
# We will create two MVNs with batch_shape = event_shape = 2.
mean = [[-1., 0], # batch:0
[0., 1]] # batch:1
chol_cov = [[[1., 0],
[0, 1]], # batch:0
[[1, 0],
[2, 2]]] # batch:1
mvn1 = ds.TransformedDistribution(
distribution=ds.Normal(loc=0., scale=1.),
bijector=ds.bijectors.Affine(shift=mean, scale_tril=chol_cov),
batch_shape=[2], # Valid because base_distribution.batch_shape == [].
event_shape=[2]) # Valid because base_distribution.event_shape == [].
mvn2 = ds.MultivariateNormalTriL(loc=mean, scale_tril=chol_cov)
# mvn1.log_prob(x) == mvn2.log_prob(x)
```
"""
@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,
distribution,
bijector=None,
batch_shape=None,
event_shape=None,
validate_args=False,
name=None):
"""Construct a Transformed Distribution.
Args:
distribution: The base distribution instance to transform. Typically an
instance of `Distribution`.
bijector: The object responsible for calculating the transformation.
Typically an instance of `Bijector`. `None` means `Identity()`.
batch_shape: `integer` vector `Tensor` which overrides `distribution`
`batch_shape`; valid only if `distribution.is_scalar_batch()`.
event_shape: `integer` vector `Tensor` which overrides `distribution`
`event_shape`; valid only if `distribution.is_scalar_event()`.
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.
name: Python `str` name prefixed to Ops created by this class. Default:
`bijector.name + distribution.name`.
"""
parameters = dict(locals())
name = name or (("" if bijector is None else bijector.name) +
distribution.name)
with ops.name_scope(name, values=[event_shape, batch_shape]) as name:
# For convenience we define some handy constants.
self._zero = constant_op.constant(0, dtype=dtypes.int32, name="zero")
self._empty = constant_op.constant([], dtype=dtypes.int32, name="empty")
if bijector is None:
bijector = identity_bijector.Identity(validate_args=validate_args)
# We will keep track of a static and dynamic version of
# self._is_{batch,event}_override. This way we can do more prior to graph
# execution, including possibly raising Python exceptions.
self._override_batch_shape = self._maybe_validate_shape_override(
batch_shape, distribution.is_scalar_batch(), validate_args,
"batch_shape")
self._is_batch_override = _logical_not(_logical_equal(
_ndims_from_shape(self._override_batch_shape), self._zero))
self._is_maybe_batch_override = bool(
tensor_util.constant_value(self._override_batch_shape) is None or
tensor_util.constant_value(self._override_batch_shape).size != 0)
self._override_event_shape = self._maybe_validate_shape_override(
event_shape, distribution.is_scalar_event(), validate_args,
"event_shape")
self._is_event_override = _logical_not(_logical_equal(
_ndims_from_shape(self._override_event_shape), self._zero))
self._is_maybe_event_override = bool(
tensor_util.constant_value(self._override_event_shape) is None or
tensor_util.constant_value(self._override_event_shape).size != 0)
# To convert a scalar distribution into a multivariate distribution we
# will draw dims from the sample dims, which are otherwise iid. This is
# easy to do except in the case that the base distribution has batch dims
# and we're overriding event shape. When that case happens the event dims
# will incorrectly be to the left of the batch dims. In this case we'll
# cyclically permute left the new dims.
self._needs_rotation = _logical_and(
self._is_event_override,
_logical_not(self._is_batch_override),
_logical_not(distribution.is_scalar_batch()))
override_event_ndims = _ndims_from_shape(self._override_event_shape)
self._rotate_ndims = _pick_scalar_condition(
self._needs_rotation, override_event_ndims, 0)
# We'll be reducing the head dims (if at all), i.e., this will be []
# if we don't need to reduce.
self._reduce_event_indices = math_ops.range(
self._rotate_ndims - override_event_ndims, self._rotate_ndims)
self._distribution = distribution
self._bijector = bijector
super(TransformedDistribution, self).__init__(
dtype=self._distribution.dtype,
reparameterization_type=self._distribution.reparameterization_type,
validate_args=validate_args,
allow_nan_stats=self._distribution.allow_nan_stats,
parameters=parameters,
# We let TransformedDistribution access _graph_parents since this class
# is more like a baseclass than derived.
graph_parents=(distribution._graph_parents + # pylint: disable=protected-access
bijector.graph_parents),
name=name)
@property
def distribution(self):
"""Base distribution, p(x)."""
return self._distribution
@property
def bijector(self):
"""Function transforming x => y."""
return self._bijector
def _event_shape_tensor(self):
return self.bijector.forward_event_shape_tensor(
distribution_util.pick_vector(
self._is_event_override,
self._override_event_shape,
self.distribution.event_shape_tensor()))
def _event_shape(self):
# If there's a chance that the event_shape has been overridden, we return
# what we statically know about the `event_shape_override`. This works
# because: `_is_maybe_event_override` means `static_override` is `None` or a
# non-empty list, i.e., we don't statically know the `event_shape` or we do.
#
# Since the `bijector` may change the `event_shape`, we then forward what we
# know to the bijector. This allows the `bijector` to have final say in the
# `event_shape`.
static_override = tensor_util.constant_value_as_shape(
self._override_event_shape)
return self.bijector.forward_event_shape(
static_override
if self._is_maybe_event_override
else self.distribution.event_shape)
def _batch_shape_tensor(self):
return distribution_util.pick_vector(
self._is_batch_override,
self._override_batch_shape,
self.distribution.batch_shape_tensor())
def _batch_shape(self):
# If there's a chance that the batch_shape has been overridden, we return
# what we statically know about the `batch_shape_override`. This works
# because: `_is_maybe_batch_override` means `static_override` is `None` or a
# non-empty list, i.e., we don't statically know the `batch_shape` or we do.
#
# Notice that this implementation parallels the `_event_shape` except that
# the `bijector` doesn't get to alter the `batch_shape`. Recall that
# `batch_shape` is a property of a distribution while `event_shape` is
# shared between both the `distribution` instance and the `bijector`.
static_override = tensor_util.constant_value_as_shape(
self._override_batch_shape)
return (static_override
if self._is_maybe_batch_override
else self.distribution.batch_shape)
def _sample_n(self, n, seed=None):
sample_shape = _concat_vectors(
distribution_util.pick_vector(self._needs_rotation, self._empty, [n]),
self._override_batch_shape,
self._override_event_shape,
distribution_util.pick_vector(self._needs_rotation, [n], self._empty))
x = self.distribution.sample(sample_shape=sample_shape, seed=seed)
x = self._maybe_rotate_dims(x)
# We'll apply the bijector in the `_call_sample_n` function.
return x
def _call_sample_n(self, sample_shape, seed, name, **kwargs):
# We override `_call_sample_n` rather than `_sample_n` so we can ensure that
# the result of `self.bijector.forward` is not modified (and thus caching
# works).
with self._name_scope(name, values=[sample_shape]):
sample_shape = ops.convert_to_tensor(
sample_shape, dtype=dtypes.int32, name="sample_shape")
sample_shape, n = self._expand_sample_shape_to_vector(
sample_shape, "sample_shape")
# First, generate samples. We will possibly generate extra samples in the
# event that we need to reinterpret the samples as part of the
# event_shape.
x = self._sample_n(n, seed, **kwargs)
# Next, we reshape `x` into its final form. We do this prior to the call
# to the bijector to ensure that the bijector caching works.
batch_event_shape = array_ops.shape(x)[1:]
final_shape = array_ops.concat([sample_shape, batch_event_shape], 0)
x = array_ops.reshape(x, final_shape)
# Finally, we apply the bijector's forward transformation. For caching to
# work, it is imperative that this is the last modification to the
# returned result.
y = self.bijector.forward(x, **kwargs)
y = self._set_sample_static_shape(y, sample_shape)
return y
def _log_prob(self, y):
# For caching to work, it is imperative that the bijector is the first to
# modify the input.
x = self.bijector.inverse(y)
event_ndims = self._maybe_get_static_event_ndims()
ildj = self.bijector.inverse_log_det_jacobian(y, event_ndims=event_ndims)
if self.bijector._is_injective: # pylint: disable=protected-access
return self._finish_log_prob_for_one_fiber(y, x, ildj, event_ndims)
lp_on_fibers = [
self._finish_log_prob_for_one_fiber(y, x_i, ildj_i, event_ndims)
for x_i, ildj_i in zip(x, ildj)]
return math_ops.reduce_logsumexp(
array_ops_stack.stack(lp_on_fibers), axis=0)
def _finish_log_prob_for_one_fiber(self, y, x, ildj, event_ndims):
"""Finish computation of log_prob on one element of the inverse image."""
x = self._maybe_rotate_dims(x, rotate_right=True)
log_prob = self.distribution.log_prob(x)
if self._is_maybe_event_override:
log_prob = math_ops.reduce_sum(log_prob, self._reduce_event_indices)
log_prob += math_ops.cast(ildj, log_prob.dtype)
if self._is_maybe_event_override and isinstance(event_ndims, int):
log_prob.set_shape(
array_ops.broadcast_static_shape(
y.get_shape().with_rank_at_least(1)[:-event_ndims],
self.batch_shape))
return log_prob
def _prob(self, y):
x = self.bijector.inverse(y)
event_ndims = self._maybe_get_static_event_ndims()
ildj = self.bijector.inverse_log_det_jacobian(y, event_ndims=event_ndims)
if self.bijector._is_injective: # pylint: disable=protected-access
return self._finish_prob_for_one_fiber(y, x, ildj, event_ndims)
prob_on_fibers = [
self._finish_prob_for_one_fiber(y, x_i, ildj_i, event_ndims)
for x_i, ildj_i in zip(x, ildj)]
return sum(prob_on_fibers)
def _finish_prob_for_one_fiber(self, y, x, ildj, event_ndims):
"""Finish computation of prob on one element of the inverse image."""
x = self._maybe_rotate_dims(x, rotate_right=True)
prob = self.distribution.prob(x)
if self._is_maybe_event_override:
prob = math_ops.reduce_prod(prob, self._reduce_event_indices)
prob *= math_ops.exp(math_ops.cast(ildj, prob.dtype))
if self._is_maybe_event_override and isinstance(event_ndims, int):
prob.set_shape(
array_ops.broadcast_static_shape(
y.get_shape().with_rank_at_least(1)[:-event_ndims],
self.batch_shape))
return prob
def _log_cdf(self, y):
if self._is_maybe_event_override:
raise NotImplementedError("log_cdf is not implemented when overriding "
"event_shape")
if not self.bijector._is_injective: # pylint: disable=protected-access
raise NotImplementedError("log_cdf is not implemented when "
"bijector is not injective.")
x = self.bijector.inverse(y)
return self.distribution.log_cdf(x)
def _cdf(self, y):
if self._is_maybe_event_override:
raise NotImplementedError("cdf is not implemented when overriding "
"event_shape")
if not self.bijector._is_injective: # pylint: disable=protected-access
raise NotImplementedError("cdf is not implemented when "
"bijector is not injective.")
x = self.bijector.inverse(y)
return self.distribution.cdf(x)
def _log_survival_function(self, y):
if self._is_maybe_event_override:
raise NotImplementedError("log_survival_function is not implemented when "
"overriding event_shape")
if not self.bijector._is_injective: # pylint: disable=protected-access
raise NotImplementedError("log_survival_function is not implemented when "
"bijector is not injective.")
x = self.bijector.inverse(y)
return self.distribution.log_survival_function(x)
def _survival_function(self, y):
if self._is_maybe_event_override:
raise NotImplementedError("survival_function is not implemented when "
"overriding event_shape")
if not self.bijector._is_injective: # pylint: disable=protected-access
raise NotImplementedError("survival_function is not implemented when "
"bijector is not injective.")
x = self.bijector.inverse(y)
return self.distribution.survival_function(x)
def _quantile(self, value):
if self._is_maybe_event_override:
raise NotImplementedError("quantile is not implemented when overriding "
"event_shape")
if not self.bijector._is_injective: # pylint: disable=protected-access
raise NotImplementedError("quantile is not implemented when "
"bijector is not injective.")
# x_q is the "qth quantile" of X iff q = P[X <= x_q]. Now, since X =
# g^{-1}(Y), q = P[X <= x_q] = P[g^{-1}(Y) <= x_q] = P[Y <= g(x_q)],
# implies the qth quantile of Y is g(x_q).
inv_cdf = self.distribution.quantile(value)
return self.bijector.forward(inv_cdf)
def _entropy(self):
if not self.bijector.is_constant_jacobian:
raise NotImplementedError("entropy is not implemented")
if not self.bijector._is_injective: # pylint: disable=protected-access
raise NotImplementedError("entropy is not implemented when "
"bijector is not injective.")
# Suppose Y = g(X) where g is a diffeomorphism and X is a continuous rv. It
# can be shown that:
# H[Y] = H[X] + E_X[(log o abs o det o J o g)(X)].
# If is_constant_jacobian then:
# E_X[(log o abs o det o J o g)(X)] = (log o abs o det o J o g)(c)
# where c can by anything.
entropy = self.distribution.entropy()
if self._is_maybe_event_override:
# H[X] = sum_i H[X_i] if X_i are mutually independent.
# This means that a reduce_sum is a simple rescaling.
entropy *= math_ops.cast(math_ops.reduce_prod(self._override_event_shape),
dtype=entropy.dtype.base_dtype)
if self._is_maybe_batch_override:
new_shape = array_ops.concat([
_ones_like(self._override_batch_shape),
self.distribution.batch_shape_tensor()
], 0)
entropy = array_ops.reshape(entropy, new_shape)
multiples = array_ops.concat([
self._override_batch_shape,
_ones_like(self.distribution.batch_shape_tensor())
], 0)
entropy = array_ops.tile(entropy, multiples)
dummy = array_ops.zeros(
shape=array_ops.concat(
[self.batch_shape_tensor(), self.event_shape_tensor()],
0),
dtype=self.dtype)
event_ndims = (self.event_shape.ndims if self.event_shape.ndims is not None
else array_ops.size(self.event_shape_tensor()))
ildj = self.bijector.inverse_log_det_jacobian(
dummy, event_ndims=event_ndims)
entropy -= math_ops.cast(ildj, entropy.dtype)
entropy.set_shape(self.batch_shape)
return entropy
def _maybe_validate_shape_override(self, override_shape, base_is_scalar,
validate_args, name):
"""Helper to __init__ which ensures override batch/event_shape are valid."""
if override_shape is None:
override_shape = []
override_shape = ops.convert_to_tensor(override_shape, dtype=dtypes.int32,
name=name)
if not override_shape.dtype.is_integer:
raise TypeError("shape override must be an integer")
override_is_scalar = _is_scalar_from_shape(override_shape)
if tensor_util.constant_value(override_is_scalar):
return self._empty
dynamic_assertions = []
if override_shape.get_shape().ndims is not None:
if override_shape.get_shape().ndims != 1:
raise ValueError("shape override must be a vector")
elif validate_args:
dynamic_assertions += [check_ops.assert_rank(
override_shape, 1,
message="shape override must be a vector")]
if tensor_util.constant_value(override_shape) is not None:
if any(s <= 0 for s in tensor_util.constant_value(override_shape)):
raise ValueError("shape override must have positive elements")
elif validate_args:
dynamic_assertions += [check_ops.assert_positive(
override_shape,
message="shape override must have positive elements")]
is_both_nonscalar = _logical_and(_logical_not(base_is_scalar),
_logical_not(override_is_scalar))
if tensor_util.constant_value(is_both_nonscalar) is not None:
if tensor_util.constant_value(is_both_nonscalar):
raise ValueError("base distribution not scalar")
elif validate_args:
dynamic_assertions += [check_ops.assert_equal(
is_both_nonscalar, False,
message="base distribution not scalar")]
if not dynamic_assertions:
return override_shape
return control_flow_ops.with_dependencies(
dynamic_assertions, override_shape)
def _maybe_rotate_dims(self, x, rotate_right=False):
"""Helper which rolls left event_dims left or right event_dims right."""
needs_rotation_const = tensor_util.constant_value(self._needs_rotation)
if needs_rotation_const is not None and not needs_rotation_const:
return x
ndims = array_ops.rank(x)
n = (ndims - self._rotate_ndims) if rotate_right else self._rotate_ndims
return array_ops.transpose(
x, _concat_vectors(math_ops.range(n, ndims), math_ops.range(0, n)))
def _maybe_get_static_event_ndims(self):
if self.event_shape.ndims is not None:
return self.event_shape.ndims
event_ndims = array_ops.size(self.event_shape_tensor())
event_ndims_ = distribution_util.maybe_get_static_value(event_ndims)
if event_ndims_ is not None:
return event_ndims_
return event_ndims
tensorflow/python/ops/distributions/uniform.py
+204
View File
@@ -0,0 +1,204 @@
# 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
# limitations under the License.
# ==============================================================================
"""The Uniform distribution class."""
import math
from tensorflow.python.framework import constant_op
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.framework import tensor_shape
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import check_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops import random_ops
from tensorflow.python.ops.distributions import distribution
from tensorflow.python.util import deprecation
from tensorflow.python.util.tf_export import tf_export
@tf_export(v1=["distributions.Uniform"])
class Uniform(distribution.Distribution):
"""Uniform distribution with `low` and `high` parameters.
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; a, b) = I[a <= x < b] / Z
Z = b - a
```
where
- `low = a`,
- `high = b`,
- `Z` is the normalizing constant, and
- `I[predicate]` is the [indicator function](
https://en.wikipedia.org/wiki/Indicator_function) for `predicate`.
The parameters `low` and `high` must be shaped in a way that supports
broadcasting (e.g., `high - low` is a valid operation).
#### Examples
```python
# Without broadcasting:
u1 = Uniform(low=3.0, high=4.0) # a single uniform distribution [3, 4]
u2 = Uniform(low=[1.0, 2.0],
high=[3.0, 4.0]) # 2 distributions [1, 3], [2, 4]
u3 = Uniform(low=[[1.0, 2.0],
[3.0, 4.0]],
high=[[1.5, 2.5],
[3.5, 4.5]]) # 4 distributions
```
```python
# With broadcasting:
u1 = Uniform(low=3.0, high=[5.0, 6.0, 7.0]) # 3 distributions
```
"""
@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,
low=0.,
high=1.,
validate_args=False,
allow_nan_stats=True,
name="Uniform"):
"""Initialize a batch of Uniform distributions.
Args:
low: Floating point tensor, lower boundary of the output interval. Must
have `low < high`.
high: Floating point tensor, upper boundary of the output interval. Must
have `low < high`.
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:
InvalidArgumentError: if `low >= high` and `validate_args=False`.
"""
parameters = dict(locals())
with ops.name_scope(name, values=[low, high]) as name:
with ops.control_dependencies([
check_ops.assert_less(
low, high, message="uniform not defined when low >= high.")
] if validate_args else []):
self._low = array_ops.identity(low, name="low")
self._high = array_ops.identity(high, name="high")
check_ops.assert_same_float_dtype([self._low, self._high])
super(Uniform, self).__init__(
dtype=self._low.dtype,
reparameterization_type=distribution.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
graph_parents=[self._low,
self._high],
name=name)
@staticmethod
def _param_shapes(sample_shape):
return dict(
zip(("low", "high"),
([ops.convert_to_tensor(sample_shape, dtype=dtypes.int32)] * 2)))
@property
def low(self):
"""Lower boundary of the output interval."""
return self._low
@property
def high(self):
"""Upper boundary of the output interval."""
return self._high
def range(self, name="range"):
"""`high - low`."""
with self._name_scope(name):
return self.high - self.low
def _batch_shape_tensor(self):
return array_ops.broadcast_dynamic_shape(
array_ops.shape(self.low),
array_ops.shape(self.high))
def _batch_shape(self):
return array_ops.broadcast_static_shape(
self.low.get_shape(),
self.high.get_shape())
def _event_shape_tensor(self):
return constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.TensorShape([])
def _sample_n(self, n, seed=None):
shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
samples = random_ops.random_uniform(shape=shape,
dtype=self.dtype,
seed=seed)
return self.low + self.range() * samples
def _prob(self, x):
broadcasted_x = x * array_ops.ones(
self.batch_shape_tensor(), dtype=x.dtype)
return array_ops.where_v2(
math_ops.is_nan(broadcasted_x), broadcasted_x,
array_ops.where_v2(
math_ops.logical_or(broadcasted_x < self.low,
broadcasted_x >= self.high),
array_ops.zeros_like(broadcasted_x),
array_ops.ones_like(broadcasted_x) / self.range()))
def _cdf(self, x):
broadcast_shape = array_ops.broadcast_dynamic_shape(
array_ops.shape(x), self.batch_shape_tensor())
zeros = array_ops.zeros(broadcast_shape, dtype=self.dtype)
ones = array_ops.ones(broadcast_shape, dtype=self.dtype)
broadcasted_x = x * ones
result_if_not_big = array_ops.where_v2(
x < self.low, zeros, (broadcasted_x - self.low) / self.range())
return array_ops.where_v2(x >= self.high, ones, result_if_not_big)
def _entropy(self):
return math_ops.log(self.range())
def _mean(self):
return (self.low + self.high) / 2.
def _variance(self):
return math_ops.square(self.range()) / 12.
def _stddev(self):
return self.range() / math.sqrt(12.)
tensorflow/python/ops/distributions/util.py
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