Consistent estimation of the basic neighborhood of Markov random fields

Details Consistent estimation of the basic neighborhood of Markov random fields Consistent estimation of the basic neighborhood of Markov random fields

2006-05-12

arxiv.org/abs/math/0605323v2

Annals of Statistics 2006, Vol. 34, No. 1, 123-145

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053605000000912 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053605000000912

For Markov random fields on $\mathbb{Z}^d$ with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the Bayesian Information Criterion, replacing likelihood by pseudo-likelihood, is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions: the estimated basic neighborhood equals the true one eventually almost surely, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results.

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