# Hammersley–Clifford theorem

The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics, that gives necessary and sufficient conditions under which a positive probability distribution can be represented as a Markov network (also known as a Markov random field). It states that a probability distribution that has a positive mass or density satisfies one of the Markov properties with respect to an undirected graph G if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph.

The relationship between Markov and Gibbs random fields was initiated by Roland Dobrushin[1] and Frank Spitzer[2] in the context of statistical mechanics. The theorem is named after John Hammersley and Peter Clifford who proved the equivalence in an unpublished paper in 1971.[3][4] Simpler proofs using the inclusion-exclusion principle were given independently by Geoffrey Grimmett,[5] Preston[6] and Sherman[7] in 1973, with a further proof by Julian Besag in 1974.[8]

## Notes

1. ^ Dobrushin, P. L. (1968), "The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity", Theory of Probability and its Applications 13 (2): 197–224, doi:10.1137/1113026
2. ^ Spitzer, Frank (1971), "Markov Random Fields and Gibbs Ensembles", The American Mathematical Monthly 78 (2): 142–154, doi:10.2307/2317621, JSTOR 2317621
3. ^ Hammersley, J. M.; Clifford, P. (1971), Markov fields on finite graphs and lattices
4. ^ Clifford, P. (1990), "Markov random fields in statistics", in Grimmett, G.R.; Welsh, D.J.A., Disorder in Physical Systems: A Volume in Honour of John M. Hammersley, Oxford University Press, pp. 19–32, ISBN 0-19-853215-6, MR 1064553, retrieved 2009-05-04
5. ^ Grimmett, G. R. (1973), "A theorem about random fields", Bulletin of the London Mathematical Society 5 (1): 81–84, doi:10.1112/blms/5.1.81, MR 0329039
6. ^ Preston, C. J. (1973), "Generalized Gibbs states and Markov random fields", Advances in Applied Probability 5 (2): 242–261, doi:10.2307/1426035, JSTOR 1426035, MR 0405645
7. ^ Sherman, S. (1973), "Markov random fields and Gibbs random fields", Israel Journal of Mathematics 14 (1): 92–103, doi:10.1007/BF02761538, MR 0321185
8. ^ Besag, J. (1974), "Spatial interaction and the statistical analysis of lattice systems", Journal of the Royal Statistical Society, Series B 36 (2): 192–236, JSTOR 2984812, MR 0373208