# Copyright 2016-2020 The GPflow Contributors. 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,
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# See the License for the specific language governing permissions and
# limitations under the License.
import numpy as np
import tensorflow as tf
from .base import TensorType
from .utilities import to_default_float
[docs]def gaussian(x: TensorType, mu: TensorType, var: TensorType) -> tf.Tensor:
return -0.5 * (np.log(2 * np.pi) + tf.math.log(var) + tf.square(mu - x) / var)
[docs]def lognormal(x: TensorType, mu: TensorType, var: TensorType) -> tf.Tensor:
lnx = tf.math.log(x)
return gaussian(lnx, mu, var) - lnx
[docs]def bernoulli(x: TensorType, p: TensorType) -> tf.Tensor:
return tf.math.log(tf.where(tf.equal(x, 1), p, 1 - p))
[docs]def poisson(x: TensorType, lam: TensorType) -> tf.Tensor:
return x * tf.math.log(lam) - lam - tf.math.lgamma(x + 1.0)
[docs]def exponential(x: TensorType, scale: TensorType) -> tf.Tensor:
return -x / scale - tf.math.log(scale)
[docs]def gamma(x: TensorType, shape: TensorType, scale: TensorType) -> tf.Tensor:
return (
-shape * tf.math.log(scale)
- tf.math.lgamma(shape)
+ (shape - 1.0) * tf.math.log(x)
- x / scale
)
[docs]def student_t(x: TensorType, mean: TensorType, scale: TensorType, df: TensorType) -> tf.Tensor:
df = to_default_float(df)
const = (
tf.math.lgamma((df + 1.0) * 0.5)
- tf.math.lgamma(df * 0.5)
- 0.5 * (tf.math.log(tf.square(scale)) + tf.math.log(df) + np.log(np.pi))
)
return const - 0.5 * (df + 1.0) * tf.math.log(
1.0 + (1.0 / df) * (tf.square((x - mean) / scale))
)
[docs]def beta(x: TensorType, alpha: TensorType, beta: TensorType) -> tf.Tensor:
# need to clip x, since log of 0 is nan...
x = tf.clip_by_value(x, 1e-6, 1 - 1e-6)
return (
(alpha - 1.0) * tf.math.log(x)
+ (beta - 1.0) * tf.math.log(1.0 - x)
+ tf.math.lgamma(alpha + beta)
- tf.math.lgamma(alpha)
- tf.math.lgamma(beta)
)
[docs]def laplace(x: TensorType, mu: TensorType, sigma: TensorType) -> tf.Tensor:
return -tf.abs(mu - x) / sigma - tf.math.log(2.0 * sigma)
[docs]def multivariate_normal(x: TensorType, mu: TensorType, L: TensorType) -> tf.Tensor:
"""
Computes the log-density of a multivariate normal.
:param x : Dx1 or DxN sample(s) for which we want the density
:param mu : Dx1 or DxN mean(s) of the normal distribution
:param L : DxD Cholesky decomposition of the covariance matrix
:return p : (1,) or (N,) vector of log densities for each of the N x's and/or mu's
x and mu are either vectors or matrices. If both are vectors (N,1):
p[0] = log pdf(x) where x ~ N(mu, LL^T)
If at least one is a matrix, we assume independence over the *columns*:
the number of rows must match the size of L. Broadcasting behaviour:
p[n] = log pdf of:
x[n] ~ N(mu, LL^T) or x ~ N(mu[n], LL^T) or x[n] ~ N(mu[n], LL^T)
"""
d = x - mu
alpha = tf.linalg.triangular_solve(L, d, lower=True)
num_dims = tf.cast(tf.shape(d)[0], L.dtype)
p = -0.5 * tf.reduce_sum(tf.square(alpha), 0)
p -= 0.5 * num_dims * np.log(2 * np.pi)
p -= tf.reduce_sum(tf.math.log(tf.linalg.diag_part(L)))
shape_constraints = [
(d, ["D", "N"]),
(L, ["D", "D"]),
(p, ["N"]),
]
tf.debugging.assert_shapes(shape_constraints, message="multivariate_normal()")
return p