Gibbs Random Fields and Markov Random Fields with Constraints

TL;DR

This paper provides a simpler proof of the equivalence between Gibbs and Markov random fields, extending it to cases with constraints and infinite fields, simplifying understanding of their relationship.

Contribution

It introduces a straightforward proof of the Gibbs-Markov equivalence that includes constrained and infinite random fields, broadening the theoretical framework.

Findings

Simplified proof of Gibbs-Markov equivalence

Extension to constrained random fields with zero probabilities

Applicability to infinite random fields

Abstract

It was shown many times in the literature that a Markov random field is equivalent to a Gibbs random field when all realizations of the field have non-zero probabilities; the proofs are rather complicated. A simpler proof, which is based directly on simple probability theory, is presented. Furthermore, it is shown that the equivalence is still valid when there are constraints (zero probability realizations) of any type. The equivalence extends to infinite size random fields, as well.