On Information-Theoretic Characterizations of Markov Random Fields and Subfields

TL;DR

This paper characterizes the minimal graph representations of subfields of Markov random fields, establishes conditions for subfields to be Markov trees, and develops recursive methods for constructing information diagrams, highlighting the uniqueness of Markov chains.

Contribution

It provides a novel set-theoretic framework for minimal graph representations of MRF subfields and characterizes when subfields are also Markov trees, advancing understanding of information diagrams.

Findings

Minimal graph representations for subfields of MRFs identified

Necessary and sufficient conditions for subfields to be Markov trees established

Recursive methods developed for constructing information diagrams

Abstract

Let $X_i, i \in V$ form a Markov random field (MRF) represented by an undirected graph $G = (V,E)$, and $V'$ be a subset of $V$. We determine the smallest graph that can always represent the subfield $X_i, i \in V'$ as an MRF. Based on this result, we obtain a necessary and sufficient condition for a subfield of a Markov tree to be also a Markov tree. When $G$ is a path so that $X_i, i \in V$ form a Markov chain, it is known that the $I$-Measure is always nonnegative and the information diagram assumes a very special structure Kawabata and Yeung (1992). We prove that Markov chain is essentially the only MRF such that the $I$-Measure is always nonnegative. By applying our characterization of the smallest graph representation of a subfield of an MRF, we develop a recursive approach for constructing information diagrams for MRFs. Our work is built on the set-theoretic characterization of an MRF in Yeung, Lee, and Ye (2002).