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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."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
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
__all__ = [
"Laplace",
"LaplaceWithSoftplusScale",
]
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
```
"""
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 = 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(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.scalar()
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`."""
def __init__(self,
loc,
scale,
validate_args=False,
allow_nan_stats=True,
name="LaplaceWithSoftplusScale"):
parameters = locals()
with ops.name_scope(name, values=[loc, scale]):
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
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