Factorization of Discrete Probability Distributions

TL;DR

This paper establishes comprehensive conditions under which discrete probability distributions can be factorized according to various graphical and exponential models, extending the classical Hammersley-Clifford Theorem.

Contribution

It generalizes the Hammersley-Clifford Theorem by providing necessary and sufficient conditions for factorization in broader classes of models.

Findings

Derived conditions for distribution factorization in exponential models

Extended the Hammersley-Clifford Theorem to more general settings

Provided a unified framework for understanding distribution factorization

Abstract

We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. This result generalizes the well known Hammersley-Clifford Theorem.