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

Abstract

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 and Markov 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. Personal acknowledgment. One of us (Gandolfi) learned about the theory of Gibbs and Markov random fields from Lucio Russo in a course based on [Ruelle 1978]. He is indebted to Lucio for his inspirational lectures and for many other things, such as an interest in percolation theory and statistical physics, a deep conviction of the close relation between abstract and applied mathematics, and an involvement in questions about the history of science. This paper focuses on one instance of this close association between abstract and applied mathematics, namely the relation between Gibbs and Markov random fields; in spite of the great number of studies and applications of these models, this relationship has not been appropriately investigated in the literature.

Markov Property in Generative Classifiers

“…For G-Markov classifiers the following result similar to the Hammersley-Clifford theorem (Hammersley and Clifford, 1971;Grimmett, 1973;Gandolfi and Lenarda, 2017) holds. The proof of Theorem 1 is partly based on the Möebius inversion lemma (Rota, 1987) as in the proof of the Hammersley-Clifford theorem (Grimmett, 1973;Lauritzen, 1996).…”

On the characterization of a finite random field by conditional distribution and its Gibbs form

“…Note that the Gibbs representation is especially suitable when the interaction potential has a simple form, while the application of the Möbius formula leads to a very complex expression for it and may have only theoretical interest. In addition, as noted by Gandolfi and Lenarda [15], when applying the Möbius formula, the explicit dependence of the potential on the values of the spins of physical systems is lost.…”

“…Let now ∆ 1 = {∆ z t , z ∈ X Λ\{t} , t ∈ Λ} be a one-point transition energy field which elements satisfy conditions (15), and let P Λ be a corresponding to it random field with one-point conditional distribution Q 1 (P Λ ). Then for all t ∈ Λ and x, u ∈ X t , z ∈ X ∂t , y, v ∈ X Λ\({t}∪∂t) , we have…”

Random-cluster correlation inequalities for Gibbs fields

“…We then consider an interaction φ defined on ∪ b∈B Ω b ; to include possible constraints we allow φ = ∞; although it would be more expressive to use a different collection of hyperbonds from B for possible constraints (see [GL16])), such a distinction is not needed for our purposes here.…”