On Functions of Markov Random Fields

TL;DR

This paper establishes two sufficient conditions, one information-theoretic and one based on potential functions, for a function of a Markov random field to also be an MRF on the same graph, with practical implications.

Contribution

It introduces new criteria for functions of MRFs to preserve the Markov property, linking information theory and potential functions, with practical examples.

Findings

Two sufficient conditions for functions of MRFs to be MRFs.

Illustrations of conditions through practical examples.

Partial characterization of information-preserving functions of MRFs.

Abstract

We derive two sufficient conditions for a function of a Markov random field (MRF) on a given graph to be a MRF on the same graph. The first condition is information-theoretic and parallels a recent information-theoretic characterization of lumpability of Markov chains. The second condition, which is easier to check, is based on the potential functions of the corresponding Gibbs field. We illustrate our sufficient conditions at the hand of several examples and discuss implications for practical applications of MRFs. As a side result, we give a partial characterization of functions of MRFs that are information-preserving.