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normal.py
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normal.py
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# Copyright 2018 The TensorFlow Probability Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ============================================================================
"""The Normal (Gaussian) distribution class."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
# Dependency imports
import numpy as np
import tensorflow.compat.v2 as tf
from tensorflow_probability.python.bijectors import identity as identity_bijector
from tensorflow_probability.python.bijectors import softplus as softplus_bijector
from tensorflow_probability.python.distributions import distribution
from tensorflow_probability.python.distributions import kullback_leibler
from tensorflow_probability.python.internal import assert_util
from tensorflow_probability.python.internal import dtype_util
from tensorflow_probability.python.internal import parameter_properties
from tensorflow_probability.python.internal import prefer_static as ps
from tensorflow_probability.python.internal import reparameterization
from tensorflow_probability.python.internal import samplers
from tensorflow_probability.python.internal import special_math
from tensorflow_probability.python.internal import tensor_util
__all__ = [
'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)
```
"""
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 tf.name_scope(name) as name:
dtype = dtype_util.common_dtype([loc, scale], dtype_hint=tf.float32)
self._loc = tensor_util.convert_nonref_to_tensor(
loc, dtype=dtype, name='loc')
self._scale = tensor_util.convert_nonref_to_tensor(
scale, dtype=dtype, name='scale')
super(Normal, self).__init__(
dtype=dtype,
reparameterization_type=reparameterization.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
name=name)
@classmethod
def _parameter_properties(cls, dtype, num_classes=None):
# pylint: disable=g-long-lambda
return dict(
loc=parameter_properties.ParameterProperties(),
scale=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))
# pylint: enable=g-long-lambda
@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, loc=None, scale=None):
return ps.broadcast_shape(
ps.shape(self.loc if loc is None else loc),
ps.shape(self.scale if scale is None else scale))
def _batch_shape(self):
return tf.broadcast_static_shape(self.loc.shape, self.scale.shape)
def _event_shape_tensor(self):
return tf.constant([], dtype=tf.int32)
def _event_shape(self):
return tf.TensorShape([])
def _sample_n(self, n, seed=None):
loc = tf.convert_to_tensor(self.loc)
scale = tf.convert_to_tensor(self.scale)
shape = ps.concat([[n], self._batch_shape_tensor(loc=loc, scale=scale)],
axis=0)
sampled = samplers.normal(
shape=shape, mean=0., stddev=1., dtype=self.dtype, seed=seed)
return sampled * scale + loc
def _log_prob(self, x):
scale = tf.convert_to_tensor(self.scale)
log_unnormalized = -0.5 * tf.math.squared_difference(
x / scale, self.loc / scale)
log_normalization = tf.constant(
0.5 * np.log(2. * np.pi), dtype=self.dtype) + tf.math.log(scale)
return log_unnormalized - 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 _entropy(self):
log_normalization = tf.constant(
0.5 * np.log(2. * np.pi), dtype=self.dtype) + tf.math.log(self.scale)
entropy = 0.5 + log_normalization
return entropy * tf.ones_like(self.loc)
def _mean(self):
return self.loc * tf.ones_like(self.scale)
def _quantile(self, p):
return tf.math.ndtri(p) * self.scale + self.loc
def _stddev(self):
return self.scale * tf.ones_like(self.loc)
_mode = _mean
def _z(self, x, scale=None):
"""Standardize input `x` to a unit normal."""
with tf.name_scope('standardize'):
return (x - self.loc) / (self.scale if scale is None else scale)
def _default_event_space_bijector(self):
return identity_bijector.Identity(validate_args=self.validate_args)
def _parameter_control_dependencies(self, is_init):
assertions = []
if is_init:
try:
self._batch_shape()
except ValueError:
raise ValueError(
'Arguments `loc` and `scale` must have compatible shapes; '
'loc.shape={}, scale.shape={}.'.format(
self.loc.shape, self.scale.shape))
# We don't bother checking the shapes in the dynamic case because
# all member functions access both arguments anyway.
if not self.validate_args:
assert not assertions # Should never happen.
return []
if is_init != tensor_util.is_ref(self.scale):
assertions.append(assert_util.assert_positive(
self.scale, message='Argument `scale` must be positive.'))
return assertions
@kullback_leibler.RegisterKL(Normal, Normal)
def _kl_normal_normal(a, b, name=None):
"""Calculate the batched KL divergence KL(a || b) with a and b Normal.
Args:
a: instance of a Normal distribution object.
b: instance of a Normal distribution object.
name: Name to use for created operations.
Default value: `None` (i.e., `'kl_normal_normal'`).
Returns:
kl_div: Batchwise KL(a || b)
"""
with tf.name_scope(name or 'kl_normal_normal'):
b_scale = tf.convert_to_tensor(b.scale) # We'll read it thrice.
diff_log_scale = tf.math.log(a.scale) - tf.math.log(b_scale)
return (
0.5 * tf.math.squared_difference(a.loc / b_scale, b.loc / b_scale) +
0.5 * tf.math.expm1(2. * diff_log_scale) -
diff_log_scale)