1 # Copyright 2016 The TensorFlow Authors. All Rights Reserved. 2 # 3 # Licensed under the Apache License, Version 2.0 (the "License"); 4 # you may not use this file except in compliance with the License. 5 # You may obtain a copy of the License at 6 # 7 # http://www.apache.org/licenses/LICENSE-2.0 8 # 9 # Unless required by applicable law or agreed to in writing, software 10 # distributed under the License is distributed on an "AS IS" BASIS, 11 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 12 # See the License for the specific language governing permissions and 13 # limitations under the License. 14 # ============================================================================== 15 """The Normal distribution: conjugate posterior closed form calculations.""" 16 17 from __future__ import absolute_import 18 from __future__ import division 19 from __future__ import print_function 20 21 from tensorflow.python.ops import math_ops 22 from tensorflow.python.ops.distributions import normal 23 24 25 def normal_conjugates_known_scale_posterior(prior, scale, s, n): 26 """Posterior Normal distribution with conjugate prior on the mean. 27 28 This model assumes that `n` observations (with sum `s`) come from a 29 Normal with unknown mean `loc` (described by the Normal `prior`) 30 and known variance `scale**2`. The "known scale posterior" is 31 the distribution of the unknown `loc`. 32 33 Accepts a prior Normal distribution object, having parameters 34 `loc0` and `scale0`, as well as known `scale` values of the predictive 35 distribution(s) (also assumed Normal), 36 and statistical estimates `s` (the sum(s) of the observations) and 37 `n` (the number(s) of observations). 38 39 Returns a posterior (also Normal) distribution object, with parameters 40 `(loc', scale'**2)`, where: 41 42 ``` 43 mu ~ N(mu', sigma'**2) 44 sigma'**2 = 1/(1/sigma0**2 + n/sigma**2), 45 mu' = (mu0/sigma0**2 + s/sigma**2) * sigma'**2. 46 ``` 47 48 Distribution parameters from `prior`, as well as `scale`, `s`, and `n`. 49 will broadcast in the case of multidimensional sets of parameters. 50 51 Args: 52 prior: `Normal` object of type `dtype`: 53 the prior distribution having parameters `(loc0, scale0)`. 54 scale: tensor of type `dtype`, taking values `scale > 0`. 55 The known stddev parameter(s). 56 s: Tensor of type `dtype`. The sum(s) of observations. 57 n: Tensor of type `int`. The number(s) of observations. 58 59 Returns: 60 A new Normal posterior distribution object for the unknown observation 61 mean `loc`. 62 63 Raises: 64 TypeError: if dtype of `s` does not match `dtype`, or `prior` is not a 65 Normal object. 66 """ 67 if not isinstance(prior, normal.Normal): 68 raise TypeError("Expected prior to be an instance of type Normal") 69 70 if s.dtype != prior.dtype: 71 raise TypeError( 72 "Observation sum s.dtype does not match prior dtype: %s vs. %s" 73 % (s.dtype, prior.dtype)) 74 75 n = math_ops.cast(n, prior.dtype) 76 scale0_2 = math_ops.square(prior.scale) 77 scale_2 = math_ops.square(scale) 78 scalep_2 = 1.0/(1/scale0_2 + n/scale_2) 79 return normal.Normal( 80 loc=(prior.loc/scale0_2 + s/scale_2) * scalep_2, 81 scale=math_ops.sqrt(scalep_2)) 82 83 84 def normal_conjugates_known_scale_predictive(prior, scale, s, n): 85 """Posterior predictive Normal distribution w. conjugate prior on the mean. 86 87 This model assumes that `n` observations (with sum `s`) come from a 88 Normal with unknown mean `loc` (described by the Normal `prior`) 89 and known variance `scale**2`. The "known scale predictive" 90 is the distribution of new observations, conditioned on the existing 91 observations and our prior. 92 93 Accepts a prior Normal distribution object, having parameters 94 `loc0` and `scale0`, as well as known `scale` values of the predictive 95 distribution(s) (also assumed Normal), 96 and statistical estimates `s` (the sum(s) of the observations) and 97 `n` (the number(s) of observations). 98 99 Calculates the Normal distribution(s) `p(x | sigma**2)`: 100 101 ``` 102 p(x | sigma**2) = int N(x | mu, sigma**2)N(mu | prior.loc, prior.scale**2) dmu 103 = N(x | prior.loc, 1 / (sigma**2 + prior.scale**2)) 104 ``` 105 106 Returns the predictive posterior distribution object, with parameters 107 `(loc', scale'**2)`, where: 108 109 ``` 110 sigma_n**2 = 1/(1/sigma0**2 + n/sigma**2), 111 mu' = (mu0/sigma0**2 + s/sigma**2) * sigma_n**2. 112 sigma'**2 = sigma_n**2 + sigma**2, 113 ``` 114 115 Distribution parameters from `prior`, as well as `scale`, `s`, and `n`. 116 will broadcast in the case of multidimensional sets of parameters. 117 118 Args: 119 prior: `Normal` object of type `dtype`: 120 the prior distribution having parameters `(loc0, scale0)`. 121 scale: tensor of type `dtype`, taking values `scale > 0`. 122 The known stddev parameter(s). 123 s: Tensor of type `dtype`. The sum(s) of observations. 124 n: Tensor of type `int`. The number(s) of observations. 125 126 Returns: 127 A new Normal predictive distribution object. 128 129 Raises: 130 TypeError: if dtype of `s` does not match `dtype`, or `prior` is not a 131 Normal object. 132 """ 133 if not isinstance(prior, normal.Normal): 134 raise TypeError("Expected prior to be an instance of type Normal") 135 136 if s.dtype != prior.dtype: 137 raise TypeError( 138 "Observation sum s.dtype does not match prior dtype: %s vs. %s" 139 % (s.dtype, prior.dtype)) 140 141 n = math_ops.cast(n, prior.dtype) 142 scale0_2 = math_ops.square(prior.scale) 143 scale_2 = math_ops.square(scale) 144 scalep_2 = 1.0/(1/scale0_2 + n/scale_2) 145 return normal.Normal( 146 loc=(prior.loc/scale0_2 + s/scale_2) * scalep_2, 147 scale=math_ops.sqrt(scalep_2 + scale_2)) 148
