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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.
# ==============================================================================
"""The Cauchy distribution class."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
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 random_ops
from tensorflow.python.ops.distributions import distribution
__all__ = [
"Cauchy",
]
class Cauchy(distribution.Distribution):
"""The Cauchy distribution with location `loc` and scale `scale`.
#### Mathematical details
The probability density function (pdf) is,
```none
pdf(x; loc, scale) = 1 / (pi * scale * (1 + ((x - loc) / scale)**2))
```
where `loc` is the location, and `scale` is the scale.
The Cauchy distribution is a member of the [location-scale family](
https://en.wikipedia.org/wiki/Location-scale_family), i.e.
```none
X ~ Cauchy(loc=0, scale=1)
Y ~ Cauchy(loc=loc, scale=scale)
Y = loc + scale * X
```
#### Examples
Examples of initialization of one or a batch of distributions.
```python
# Define a single scalar Cauchy distribution.
dist = Cauchy(loc=0., scale=3.)
# Evaluate the cdf at 1, returning a scalar.
dist.cdf(1.)
# Define a batch of two scalar valued Cauchy distributions.
dist = Cauchy(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 Cauchy distributions.
# Both have median 1, but different scales.
dist = tf.contrib.distributions.Cauchy(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)
```
"""
def __init__(self,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="Cauchy"):
"""Construct Cauchy distributions with loc and and scale `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 modes of the distribution(s).
scale: Floating point tensor; the locations 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 = locals()
with ops.name_scope(name, values=[loc, scale]):
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(Cauchy, 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.shape,
self.scale.shape)
def _event_shape_tensor(self):
return constant_op.constant([], dtype=dtypes.int32)
def _event_shape(self):
return tensor_shape.scalar()
def _sample_n(self, n, seed=None):
shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
probs = random_ops.random_uniform(
shape=shape, minval=0., maxval=1., dtype=self.dtype, seed=seed)
return self._quantile(probs)
def _log_prob(self, x):
return self._log_unnormalized_prob(x) - self._log_normalization()
def _cdf(self, x):
return math_ops.atan(self._z(x)) / np.pi + 0.5
def _log_cdf(self, x):
return math_ops.log1p(2 / np.pi * math_ops.atan(self._z(x))) - np.log(2)
def _log_unnormalized_prob(self, x):
return -math_ops.log1p(math_ops.square(self._z(x)))
def _log_normalization(self):
return np.log(np.pi) + math_ops.log(self.scale)
def _entropy(self):
h = np.log(4 * np.pi) + math_ops.log(self.scale)
return h * array_ops.ones_like(self.loc)
def _quantile(self, p):
return self.loc + self.scale * math_ops.tan(np.pi * (p - 0.5))
def _mode(self):
return self.loc * array_ops.ones_like(self.scale)
def _z(self, x):
"""Standardize input `x`."""
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
def _mean(self):
if self.allow_nan_stats:
return array_ops.fill(self.batch_shape_tensor(),
self.dtype.as_numpy_dtype(np.nan))
else:
raise ValueError("`mean` is undefined for Cauchy distribution.")
def _stddev(self):
if self.allow_nan_stats:
return array_ops.fill(self.batch_shape_tensor(),
self.dtype.as_numpy_dtype(np.nan))
else:
raise ValueError("`stddev` is undefined for Cauchy distribution.")
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