Context Block Estimation for Random Fields

For random fields on the d-dimensional integer lattice with finite state space, the basic neighborhood is the smallest region around a site that determines the conditional distribution at the site given the values at all other sites. For variable basic neighborhood random fields the basic neighborhood may vary with the values at the surrounding sites, and the sets of the values in these basic neighborhoods are called context blocks. The context blocks may be infinite. Statistical estimation of the context block system of a random field from a sample, a single realization of the random field observed in a finite region, is addressed. The Optimal Likelihood Ratio (OLR) estimator is introduced. Its nearly linear computation complexity is shown and a bound on the probability of the estimation error is proved that implies strong consistency of the estimator.