On the neighborhood radius estimation in Variable-neighborhood Markov Random Fields

We consider Markov Random Fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. The formal definition of these models requires partitions of the set of configurations according to their projections on finite neighborhoods of each lattice site. Each of these projections is called a context for the site. This framework is a natural extension, to d-dimensional fields, of the notion of variable-length Markov chains introduced by Rissanen (1983) in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator when the Dobrushin uniqueness condition for the one point conditional probabilities holds. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context.