## The Annals of Probability

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

### A Decomposition Theorem for Binary Markov Random Fields

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992084

**Digital Object Identifier**

doi:10.1214/aop/1176992084

**Mathematical Reviews number (MathSciNet)**

MR893917

**Zentralblatt MATH identifier**

0626.60045

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G60: Random fields

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1176992084