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New models for Markov random fields

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Noel Cressie  and
Subhash Lele
Noel Cressie*
Affiliation:
Iowa State University
Subhash Lele*
Affiliation:
The Johns Hopkins University
*
∗ Postal address: Department of Statistics, Iowa State University, Snedecor Hall, Ames, IA 50011–1210, USA.
∗∗ Postal address: Department of Biostatistics, The Johns Hopkins University, Baltimore, MD 21218, USA.
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Abstract

The Hammersley–Clifford theorem gives the form that the joint probability density (or mass) function of a Markov random field must take. Its exponent must be a sum of functions of variables, where each function in the summand involves only those variables whose sites form a clique. From a statistical modeling point of view, it is important to establish the converse result, namely, to give the conditional probability specifications that yield a Markov random field. Besag (1974) addressed this question by developing a one-parameter exponential family of conditional probability models. In this article, we develop new models for Markov random fields by establishing sufficient conditions for the conditional probability specifications to yield a Markov random field.

Keywords

AUTO-MODELSCONDITIONAL DISTRIBUTIONSEXPONENTIAL FAMILIESHAMMERSLEY-CLIFFORD THEOREMNEIGHBORHOODS

MSC classification

Primary: 60G60: Random fields
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 877 - 884
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Research partially supported by the NSF under Grants DMS8902812 and DMS9001862.

References

Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B 36, 192–225.Google Scholar
Clifford, P. (1990) Markov random fields in statistics. In Disorder in Physical Systems, ed. Grimmett, G. R. and Welsh, D. J. A., pp. 19–32. Oxford University Press.Google Scholar
Cressie, N. (1991) Statistics for Spatial Data. Wiley, New York.Google Scholar
Everitt, B. S. and Hand, D. J. (1981) Finite Mixture Distributions. Chapman and Hall, London.Google Scholar
Hammersley, J. M. and Clifford, P. (1971) Markov fields on finite graphs and lattices. Unpublished manuscript, Oxford University.Google Scholar