The Annals of Probability

A Decomposition Theorem for Binary Markov Random Fields

Bruce Hajek and Toby Berger

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Consider a binary Markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree. Under a minimal assumption we prove a decomposition theorem to the effect that such a Markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields, one of which is white binary noise of positive weight. Said decomposition provides the information theorist with an exact expression for the per-site rate-distortion function of the random field over an interval of distortions not exceeding this weight. We mention possible implications for communication theory, probability theory and statistical physics.

Article information

Ann. Probab., Volume 15, Number 3 (1987), 1112-1125.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G60: Random fields
Secondary: 94A34: Rate-distortion theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Markov random field Gibbs random field Ising model rate-distortion function


Hajek, Bruce; Berger, Toby. A Decomposition Theorem for Binary Markov Random Fields. Ann. Probab. 15 (1987), no. 3, 1112--1125. doi:10.1214/aop/1176992084.

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