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Consistent estimation of the basic neighborhood of Markov random fields. (English) Zbl 1102.62105

Author’s abstract: For Markov random fields on \(Z^d\) with finite state space, the author addresses 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 transitions do not affect the results.

MSC:

62M40 Random fields; image analysis
62F12 Asymptotic properties of parametric estimators
60G60 Random fields
62F15 Bayesian inference
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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