Markov random fields, Markov cocycles and the 3-colored chessboard

Nishant Chandgotia, Tom Meyerovitch

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

The well-known Hammersley–Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley–Clifford Theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce “Markov cocycles”, reparametrizations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley–Clifford Theorem for a family of Markov random fields which are outside the theorem’s purview where the underlying graph is Zd. This family includes all Markov random fields whose support is the d-dimensional “3-colored chessboard”. On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.

Original languageEnglish
Pages (from-to)909-964
Number of pages56
JournalIsrael Journal of Mathematics
Volume215
Issue number2
DOIs
StatePublished - 1 Sep 2016

ASJC Scopus subject areas

  • Mathematics (all)

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