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kumaraswamy.py
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kumaraswamy.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 Kumaraswamy distribution class."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import numpy as np
from tensorflow.contrib.distributions.python.ops import bijectors
from tensorflow.contrib.distributions.python.ops import distribution_util
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 special_math_ops
from tensorflow.python.ops.distributions import distribution
from tensorflow.python.ops.distributions import transformed_distribution
from tensorflow.python.ops.distributions import uniform
from tensorflow.python.util import deprecation
__all__ = [
"Kumaraswamy",
]
_kumaraswamy_sample_note = """Note: `x` must have dtype `self.dtype` and be in
`[0, 1].` It must have a shape compatible with `self.batch_shape()`."""
@deprecation.deprecated(
"2018-10-01",
"The TensorFlow Distributions library has moved to "
"TensorFlow Probability "
"(https://github.com/tensorflow/probability). You "
"should update all references to use `tfp.distributions` "
"instead of `tf.contrib.distributions`.",
warn_once=True)
def _harmonic_number(x):
"""Compute the harmonic number from its analytic continuation.
Derivation from [here](
https://en.wikipedia.org/wiki/Digamma_function#Relation_to_harmonic_numbers)
and [Euler's constant](
https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant).
Args:
x: input float.
Returns:
z: The analytic continuation of the harmonic number for the input.
"""
one = array_ops.ones([], dtype=x.dtype)
return math_ops.digamma(x + one) - math_ops.digamma(one)
class Kumaraswamy(transformed_distribution.TransformedDistribution):
"""Kumaraswamy distribution.
The Kumaraswamy distribution is defined over the `(0, 1)` interval using
parameters
`concentration1` (aka "alpha") and `concentration0` (aka "beta"). It has a
shape similar to the Beta distribution, but is reparameterizeable.
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; alpha, beta) = alpha * beta * x**(alpha - 1) * (1 - x**alpha)**(beta -
1)
```
where:
* `concentration1 = alpha`,
* `concentration0 = beta`,
Distribution parameters are automatically broadcast in all functions; see
examples for details.
#### Examples
```python
# Create a batch of three Kumaraswamy distributions.
alpha = [1, 2, 3]
beta = [1, 2, 3]
dist = Kumaraswamy(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 = Kumaraswamy(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]
```
"""
@deprecation.deprecated(
"2018-10-01",
"The TensorFlow Distributions library has moved to "
"TensorFlow Probability "
"(https://github.com/tensorflow/probability). You "
"should update all references to use `tfp.distributions` "
"instead of `tf.contrib.distributions`.",
warn_once=True)
def __init__(self,
concentration1=None,
concentration0=None,
validate_args=False,
allow_nan_stats=True,
name="Kumaraswamy"):
"""Initialize a batch of Kumaraswamy 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.
"""
with ops.name_scope(name, values=[concentration1, concentration0]) as name:
concentration1 = ops.convert_to_tensor(
concentration1, name="concentration1")
concentration0 = ops.convert_to_tensor(
concentration0, name="concentration0")
super(Kumaraswamy, self).__init__(
distribution=uniform.Uniform(
low=array_ops.zeros([], dtype=concentration1.dtype),
high=array_ops.ones([], dtype=concentration1.dtype),
allow_nan_stats=allow_nan_stats),
bijector=bijectors.Kumaraswamy(
concentration1=concentration1, concentration0=concentration0,
validate_args=validate_args),
batch_shape=distribution_util.get_broadcast_shape(
concentration1, concentration0),
name=name)
self._reparameterization_type = distribution.FULLY_REPARAMETERIZED
@property
def concentration1(self):
"""Concentration parameter associated with a `1` outcome."""
return self.bijector.concentration1
@property
def concentration0(self):
"""Concentration parameter associated with a `0` outcome."""
return self.bijector.concentration0
def _entropy(self):
a = self.concentration1
b = self.concentration0
return (1 - 1. / a) + (
1 - 1. / b) * _harmonic_number(b) + math_ops.log(a) + math_ops.log(b)
def _moment(self, n):
"""Compute the n'th (uncentered) moment."""
total_concentration = self.concentration1 + self.concentration0
expanded_concentration1 = array_ops.ones_like(
total_concentration, dtype=self.dtype) * self.concentration1
expanded_concentration0 = array_ops.ones_like(
total_concentration, dtype=self.dtype) * self.concentration0
beta_arg0 = 1 + n / expanded_concentration1
beta_arg = array_ops.stack([beta_arg0, expanded_concentration0], -1)
log_moment = math_ops.log(expanded_concentration0) + special_math_ops.lbeta(
beta_arg)
return math_ops.exp(log_moment)
def _mean(self):
return self._moment(1)
def _variance(self):
# TODO(b/72696533): Investigate a more numerically stable version.
return self._moment(2) - math_ops.square(self._moment(1))
@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):
a = self.concentration1
b = self.concentration0
mode = ((a - 1) / (a * b - 1))**(1. / a)
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 = (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.concentration1.dtype),
self.concentration1,
message="Mode undefined for concentration1 <= 1."),
check_ops.assert_less(
array_ops.ones([], dtype=self.concentration0.dtype),
self.concentration0,
message="Mode undefined for concentration0 <= 1.")
], mode)