Vol. 4, No. 3-4, 2016

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

Alberto Gandolfi and Pietro Lenarda

Vol. 4 (2016), No. 3-4, 407–422
DOI: 10.2140/memocs.2016.4.407

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.

Gibbs distributions, Markov random fields, hard-core constraints, moments, Hammersley–Clifford, Moussouris
Mathematical Subject Classification 2010
Primary: 60J99, 82B20
Secondary: 44A60, 62B05, 62M40
Received: 14 September 2016
Accepted: 12 January 2017
Published: 13 April 2017

Communicated by Raffaele Esposito
Alberto Gandolfi
NYU Abu Dhabi
PO Box 129188
Abu Dhabi
United Arab Emirates
Dipartimento di Matematica e Informatica “Ulisse Dini”
Università di Firenze
Viale Morgagni 67/a
50134 Firenze
Pietro Lenarda
Multi-scale Analysis of Materials
IMT School for Advanced Studies Lucca
I-55100 Lucca