Gibbs Random Fields and Markov Random Fields with Constraints

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.