Sherman, S. Israel J. Math. (1973) 14: 92. doi:10.1007/BF02761538
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 onZv, and (v) at anyxεZv, 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 Zv 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 Zv is replaced by a triangular lattice.