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Generalized Gibbs states and Markov random fields

Published online by Cambridge University Press:  01 July 2016

C. J. Preston
C. J. Preston*
Affiliation:
University of Oxford
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Abstract

It is shown that the set of Markov random fields and Gibbs states with nearest neighbour potentials are the same for any finite graph. The set of Markov random fields is also shown to be the same as the equilibrium states of time-reversible birth/death processes with nearest neighbour interactions defined on the graph.

Keywords

MARKOV RANDOM FIELDGIBBS STATENEAREST NEIGHBOUR POTENTIALSBIRTH/DEATH PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 2 , August 1973 , pp. 242 - 261
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Averintsev, M. B. (1970) On a method of describing discrete parameter random fields. Problemy Peredachi Informatsii 6, 100–109.Google Scholar
[2] Bartlett, M. S. (1971) Physical nearest-neighbor models and non-linear time-series. J. Appl. Prob. 8, 222–232.Google Scholar
[3] Bartlett, M. S. (1972) Physical nearest-neighbor models and non-linear time series. II. Further discussion of approximate solutions and exact equations. J. Appl. Prob. 9, 76–86.Google Scholar
[4] Hammersley, J. M. and Clifford, P. Markov fields on finite graphs and lattices. Unpublished.Google Scholar
[5] Kendall, D. G. (1959) Unitary dilations of one-parameter semi-groups of Markov transition operators. Proc. Lond. Math. Soc. 9, 417–431.Google Scholar
[6] Spitzer, F. (1971) Random fields and interacting particle systems. Lectures given to the 1971 M. A. A. Summer Seminar. Math. Assoc, of America.Google Scholar
[7] Spitzer, F. (1971) Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78, 142–154.Google Scholar
[8] Sherman, S. (1973) Markov random fields and Gibbs random fields. Israel J. Math. 14, 92–103.Google Scholar