Examples
Product of two probability distributions over the same variable
In this example, we'll show you how to use the ExponentialFamily package to calculate the product of probability distributions that both relate to the same variable "X". This operation results in another probability distribution, also centered around the variable "X". It's essential to note that this is distinct from multiplying probability distributions for two different variables, such as "X" and "Y", which yields a joint probability distribution. Calculating the product of two probability distributions over the same variable is a crucial step in applying Bayes' rule.
\[p(X\vert D) \propto \underbrace{p(X)p(D|X)}_{\mathrm{product~of~two~distributions}}\]
To perform this operation, the ExponentialFamily library employs the prod function. This function takes a product strategy as its first argument. For instance:
using ExponentialFamily, Distributions
+Examples · ExponentialFamily.jl Examples
Product of two probability distributions over the same variable
In this example, we'll show you how to use the ExponentialFamily package to calculate the product of probability distributions that both relate to the same variable "X". This operation results in another probability distribution, also centered around the variable "X". It's essential to note that this is distinct from multiplying probability distributions for two different variables, such as "X" and "Y", which yields a joint probability distribution. Calculating the product of two probability distributions over the same variable is a crucial step in applying Bayes' rule.
\[p(X\vert D) \propto \underbrace{p(X)p(D|X)}_{\mathrm{product~of~two~distributions}}\]
To perform this operation, the ExponentialFamily library employs the prod function. This function takes a product strategy as its first argument. For instance:
using ExponentialFamily, Distributions, BayesBase
prior = Bernoulli(0.5)
likelihood = Bernoulli(0.6)
@@ -10,7 +10,7 @@
ef_posterior = prod(PreserveTypeProd(ExponentialFamilyDistribution), ef_prior, ef_likelihood)
ExponentialFamily(Bernoulli)
Or even more concisely:
prod(PreserveTypeProd(ExponentialFamilyDistribution), prior, likelihood)
ExponentialFamily(Bernoulli)
In this example, multiplying two Bernoulli distributions will always result in another Bernoulli distribution. However, this is not the case for all distributions. For instance, the product of two Laplace distributions may not yield another Laplace distribution, and representing the result in the same form might not be possible. In such cases, it's advisable to calculate the result within the exponential family domain. This is because the product of two exponential family distributions can always be represented as another exponential family distribution, as shown here:
prior = Laplace(2.0, 3.0)
likelihood = Laplace(1.0, 4.0)
-prod(PreserveTypeProd(ExponentialFamilyDistribution), prior, likelihood)
ExponentialFamily(Univariate)
Note that the result does not correspond to the Laplace distribution and returns a generic univariate ExponentialFamilyDistribution. This approach ensures consistency and compatibility, especially when dealing with a wide range of probability distributions.
Computing various useful attributes of an exponential family member
The package implements attributes of many well known exponential family members, which are defined in this table. The attributes include getbasemeasure, getsufficientstatistics, getlogpartition, getfisherinformation, and others. In general, the interface for these functions assumes a family member "tag," such as Normal or Bernoulli. Here are some examples of how to use these attributes:
using ExponentialFamily, Distributions
+prod(PreserveTypeProd(ExponentialFamilyDistribution), prior, likelihood)
ExponentialFamily(Univariate)
Note that the result does not correspond to the Laplace distribution and returns a generic univariate ExponentialFamilyDistribution. This approach ensures consistency and compatibility, especially when dealing with a wide range of probability distributions. Refer to the BayesBase for the documentation about available product strategies.
Computing various useful attributes of an exponential family member
The package implements attributes of many well known exponential family members, which are defined in this table. The attributes include getbasemeasure, getsufficientstatistics, getlogpartition, getfisherinformation, and others. In general, the interface for these functions assumes a family member "tag," such as Normal or Bernoulli. Here are some examples of how to use these attributes:
using ExponentialFamily, Distributions
# Returns a function
basemeasure_of_normal = getbasemeasure(Normal)
@@ -45,4 +45,4 @@
fisherinformation_of_gamma_in_natural_space(gamma_parameters_in_natural_space)
2×2 StaticArraysCore.SMatrix{2, 2, Float64, 4} with indices SOneTo(2)×SOneTo(2):
1.64493 2.0
- 2.0 4.0
Approximating attributes
Refer to the ExpectationApproximations.jl package for approximating various attributes of the members of the exponential family.
Settings
This document was generated with Documenter.jl version 1.1.1 on Tuesday 17 October 2023. Using Julia version 1.9.3.
Approximating attributes
Refer to the ExpectationApproximations.jl package for approximating various attributes of the members of the exponential family.