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Generalized Gibbs states and Markov random fields

  • C. J. Preston (a1)

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.

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[1] Averintsev, M. B. (1970) On a method of describing discrete parameter random fields. Problemy Peredachi Informatsii 6, 100109.
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[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, 7686.
[4] Hammersley, J. M. and Clifford, P. Markov fields on finite graphs and lattices. Unpublished.
[5] Kendall, D. G. (1959) Unitary dilations of one-parameter semi-groups of Markov transition operators. Proc. Lond. Math. Soc. 9, 417431.
[6] Spitzer, F. (1971) Random fields and interacting particle systems. Lectures given to the 1971 M. A. A. Summer Seminar. Math. Assoc, of America.
[7] Spitzer, F. (1971) Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78, 142154.
[8] Sherman, S. (1973) Markov random fields and Gibbs random fields. Israel J. Math. 14, 92103.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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