Markov random fields and gibbs random fields

Spitzer has shown that every Markov random field (MRF) is a Gibbs random field (GRF) and vice versa when (i) both are translation invariant, (ii) the MRF is of first order, and (iii) the GRF is defined by a binary, nearest neighbor potential. In both cases, the field (iv) is defined onZ ^v, and (v) at anyxεZ^v, takes on one of two states. The current paper shows that a MRF is a GRF and vice versa even when (i)−(v) are relaxed, i.e., even if one relaxes translation invariance, replaces first order bykth order, allows for many states and replaces finite domains of Z^v by arbitrary finite sets. This is achieved at the expense of using a many body rather than a pair potential, which turns out to be natural even in the classical (nearest neighbor) case when Z^v is replaced by a triangular lattice. The contents of this paper were presented in August, 1971, at a seminar of the Battelle Rencontre in Statistical Mechanics and also at a pair of seminars in December, 1971, at the Weizmann Institute of Science. Partially supported by NSF GP 7469 and a Weizmann Institute senior fellowship while on sabbatical leave from Indiana University.