A note on Gibbs and Markov random fields with constraints and their moments

This paper focuses on the relation between Gibbs and Markov random fields, one instance of the close relation between abstract and applied mathematics so often stressed by Lucio Russo in his scientific work. We start by proving a more explicit version, based on spin products, of the Hammersley-Clifford theorem, a classic result which identifies Gibbs andMarkov fields under finite energy. Then we argue that the celebrated counterexample of Moussouris, intended to show that there is no complete coincidence between Markov and Gibbs random fields in the presence of hard-core constraints, is not really such. In fact, the notion of a constrained Gibbs random field used in the example and in the subsequent literature makes the unnatural assumption that the constraints are infinite energy Gibbs interactions on the same graph. Here we consider the more natural extended version of the equivalence problem, in which constraints are more generally based on a possibly larger graph, and solve it. The bearing of the more natural approach is shown by considering identifiability of discrete random fields from support, conditional independencies and corresponding moments. In fact, by means of our previous results, we show identifiability for a large class of problems, and also examples with no identifiability. Various open questions surface along the way.