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Israel Journal of Mathematics

, Volume 14, Issue 1, pp 92–103 | Cite as

Markov random fields and gibbs random fields

  • S. Sherman
Article

Abstract

Spitzer has shown that every Markov random field (MRF) is a Gibbs random field (GRF) and vice versa when (i) both are translation invariant, (ii) the MRF is of first order, and (iii) the GRF is defined by a binary, nearest neighbor potential. In both cases, the field (iv) is defined onZ v, and (v) at anyxεZv, takes on one of two states. The current paper shows that a MRF is a GRF and vice versa even when (i)−(v) are relaxed, i.e., even if one relaxes translation invariance, replaces first order bykth order, allows for many states and replaces finite domains of Zv by arbitrary finite sets. This is achieved at the expense of using a many body rather than a pair potential, which turns out to be natural even in the classical (nearest neighbor) case when Zv is replaced by a triangular lattice.

Keywords

Conditional Probability Random Field Translation Invariance Markov Random Field Finite Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. B. Averintsev,On a method of describing discrete parameter random fields, Problemy Peradači Informacii,6 (1970) 100–109.Google Scholar
  2. 2.
    M. B. Averintsev,The description of a Markov random field by Gibbs conditional distributions, Teor. Verojatnost i Primenen17 (1972), 21–35.MathSciNetGoogle Scholar
  3. 3.
    R. L. Dobrusin,Description of a random field by means of conditional probabilities and conditions governing its regularity. Theor. Probability Appl. (English translation of Teor Verojatnost i Primenen13 (1968), 197–224.MathSciNetGoogle Scholar
  4. 4.
    J. M. Hammersely and P. Clifford, Markov fields of finite graphs and lattices, (Berkeley preprint).Google Scholar
  5. 5.
    D. Ruelle,Statistical Mechanics, Benjamin, New York, 1969.zbMATHGoogle Scholar
  6. 6.
    F. Spitzer,Markov random fleds and Gibbs ensembles. Amer. Math. Montly78 (1971), 142–154.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1973

Authors and Affiliations

  • S. Sherman
    • 1
    • 2
  1. 1.The Weizmann InstituteRehovotIsrael
  2. 2.Indiana UniversityBloomingtonU.S.A.

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