/
inverse_gamma.py
305 lines (256 loc) · 10.9 KB
/
inverse_gamma.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
# 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 InverseGamma distribution class."""
# Dependency imports
import numpy as np
import tensorflow.compat.v2 as tf
from tensorflow_probability.python.bijectors import chain as chain_bijector
from tensorflow_probability.python.bijectors import reciprocal as reciprocal_bijector
from tensorflow_probability.python.bijectors import softplus as softplus_bijector
from tensorflow_probability.python.distributions import distribution
from tensorflow_probability.python.distributions import gamma as gamma_lib
from tensorflow_probability.python.internal import assert_util
from tensorflow_probability.python.internal import distribution_util
from tensorflow_probability.python.internal import dtype_util
from tensorflow_probability.python.internal import parameter_properties
from tensorflow_probability.python.internal import reparameterization
from tensorflow_probability.python.internal import tensor_util
from tensorflow_probability.python.math import special
__all__ = [
'InverseGamma',
]
class InverseGamma(distribution.AutoCompositeTensorDistribution):
"""InverseGamma distribution.
The `InverseGamma` distribution is defined over positive real numbers using
parameters `concentration` (aka "alpha") and `scale` (aka "beta").
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; alpha, beta, x > 0) = x**(-alpha - 1) exp(-beta / x) / Z
Z = Gamma(alpha) beta**-alpha
```
where:
* `concentration = alpha`,
* `scale = beta`,
* `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 [upper incomplete Gamma function](
https://en.wikipedia.org/wiki/Incomplete_gamma_function).
The parameters can be intuited via their relationship to mean and variance
when these moments exist,
```none
mean = beta / (alpha - 1) when alpha > 1
variance = beta**2 / (alpha - 1)**2 / (alpha - 2) when alpha > 2
```
i.e., under the same conditions:
```none
alpha = mean**2 / variance + 2
beta = mean * (mean**2 / variance + 1)
```
Distribution parameters are automatically broadcast in all functions; see
examples for details.
Samples of this distribution are reparameterized (pathwise differentiable).
The derivatives are computed using the approach described in the paper
[Michael Figurnov, Shakir Mohamed, Andriy Mnih.
Implicit Reparameterization Gradients, 2018](https://arxiv.org/abs/1805.08498)
#### Examples
```python
tfd = tfp.distributions
dist = tfd.InverseGamma(concentration=3.0, scale=2.0)
dist2 = tfd.InverseGamma(concentration=[3.0, 4.0], scale=[2.0, 3.0])
```
Compute the gradients of samples w.r.t. the parameters:
```python
tfd = tfp.distributions
concentration = tf.constant(3.0)
scale = tf.constant(2.0)
dist = tfd.InverseGamma(concentration, 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, [concentration, scale])
```
"""
def __init__(self,
concentration,
scale=None,
validate_args=False,
allow_nan_stats=True,
name='InverseGamma'):
"""Construct InverseGamma with `concentration` and `scale` parameters.
The parameters `concentration` and `scale` must be shaped in a way that
supports broadcasting (e.g. `concentration + scale` is a valid operation).
Args:
concentration: Floating point tensor, the concentration params of the
distribution(s). Must contain only positive values.
scale: Floating point tensor, the 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 `scale` are different dtypes.
"""
parameters = dict(locals())
with tf.name_scope(name) as name:
dtype = dtype_util.common_dtype(
[concentration, scale], dtype_hint=tf.float32)
self._concentration = tensor_util.convert_nonref_to_tensor(
concentration, dtype=dtype, name='concentration')
self._scale = tensor_util.convert_nonref_to_tensor(
scale, dtype=dtype, name='scale')
super(InverseGamma, self).__init__(
dtype=self._concentration.dtype,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
reparameterization_type=reparameterization.FULLY_REPARAMETERIZED,
parameters=parameters,
name=name)
@classmethod
def _parameter_properties(cls, dtype, num_classes=None):
# pylint: disable=g-long-lambda
return dict(
concentration=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))),
scale=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))
# pylint: enable=g-long-lambda
@property
def concentration(self):
"""Concentration parameter."""
return self._concentration
@property
def scale(self):
"""Scale parameter."""
return self._scale
def _event_shape_tensor(self):
return tf.constant([], dtype=tf.int32)
def _event_shape(self):
return tf.TensorShape([])
@distribution_util.AppendDocstring(
"""Note: See `tf.random.gamma` docstring for sampling details and
caveats.""")
def _sample_n(self, n, seed=None):
return tf.math.exp(-gamma_lib.random_gamma(
shape=[n],
concentration=self.concentration,
rate=self.scale,
seed=seed,
log_space=True))
def _log_prob(self, x):
concentration = tf.convert_to_tensor(self.concentration)
scale = tf.convert_to_tensor(self.scale)
unnormalized_prob = -(1. + concentration) * tf.math.log(x) - scale / x
normalization = (
tf.math.lgamma(concentration) - concentration * tf.math.log(scale))
return unnormalized_prob - normalization
def _cdf(self, x):
# Note that igammac returns the upper regularized incomplete gamma
# function Q(a, x), which is what we want for the CDF.
return tf.math.igammac(self.concentration, self.scale / x)
def _quantile(self, p):
return tf.math.reciprocal(
special.igammacinv(self.concentration, p)) * self.scale
def _entropy(self):
concentration = tf.convert_to_tensor(self.concentration)
scale = tf.convert_to_tensor(self.scale)
return (concentration + tf.math.log(scale) +
tf.math.lgamma(concentration) -
((1. + concentration) * tf.math.digamma(concentration)))
@distribution_util.AppendDocstring(
"""The mean of an inverse gamma distribution is
`scale / (concentration - 1)`, when `concentration > 1`, and `NaN`
otherwise. If `self.allow_nan_stats` is `False`, an exception will be
raised rather than returning `NaN`""")
def _mean(self):
concentration = tf.convert_to_tensor(self.concentration)
scale = tf.convert_to_tensor(self.scale)
mean = scale / (concentration - 1.)
if self.allow_nan_stats:
assertions = []
else:
assertions = [assert_util.assert_less(
tf.ones([], self.dtype), concentration,
message='mean undefined when any concentration <= 1')]
with tf.control_dependencies(assertions):
return tf.where(
concentration > 1.,
mean,
dtype_util.as_numpy_dtype(self.dtype)(np.nan))
@distribution_util.AppendDocstring(
"""Variance for inverse gamma is defined only for `concentration > 2`. If
`self.allow_nan_stats` is `False`, an exception will be raised rather
than returning `NaN`.""")
def _variance(self):
concentration = tf.convert_to_tensor(self.concentration)
scale = tf.convert_to_tensor(self.scale)
var = (
tf.square(scale) / tf.square(concentration - 1.) /
(concentration - 2.))
if self.allow_nan_stats:
assertions = []
else:
assertions = [assert_util.assert_less(
tf.constant(2., dtype=self.dtype),
concentration,
message='variance undefined when any concentration <= 2')]
with tf.control_dependencies(assertions):
return tf.where(
concentration > 2.,
var,
dtype_util.as_numpy_dtype(self.dtype)(np.nan))
@distribution_util.AppendDocstring(
"""The mode of an inverse gamma distribution is `scale / (concentration +
1)`.""")
def _mode(self):
return self.scale / (1. + self.concentration)
def _default_event_space_bijector(self):
return chain_bijector.Chain([
reciprocal_bijector.Reciprocal(validate_args=self.validate_args),
softplus_bijector.Softplus(validate_args=self.validate_args)
], validate_args=self.validate_args)
def _sample_control_dependencies(self, x):
assertions = []
if not self.validate_args:
return assertions
assertions.append(assert_util.assert_non_negative(
x, message='Sample must be non-negative.'))
return assertions
def _parameter_control_dependencies(self, is_init):
if not self.validate_args:
return []
assertions = []
if is_init != tensor_util.is_ref(self.concentration):
assertions.append(assert_util.assert_positive(
self.concentration,
message='Argument `concentration` must be positive.'))
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