Partially observed Markov random fields are variable neighborhood random fields

TL;DR

This paper introduces variable neighborhood random fields as a new class of random fields, analyzing their properties in partially observed Markov random fields and establishing conditions for finite neighborhoods related to phase transitions.

Contribution

It presents the concept of variable neighborhood random fields, providing conditions for their finiteness and exploring their relation to phase transitions in Markov random fields.

Findings

Finite neighborhoods depend on noise level and specification values in non-phase transition cases.

Infinite neighborhoods are linked to phase transitions in the ferromagnetic Ising model.

Conditions for almost sure finiteness of neighborhoods are established.

Abstract

The present paper has two goals. First to present a natural example of a new class of random fields which are the variable neighborhood random fields. The example we consider is a partially observed nearest neighbor binary Markov random field. The second goal is to establish sufficient conditions ensuring that the variable neighborhoods are almost surely finite. We discuss the relationship between the almost sure finiteness of the interaction neighborhoods and the presence/absence of phase transition of the underlying Markov random field. In the case where the underlying random field has no phase transition we show that the finiteness of neighborhoods depends on a specific relation between the noise level and the minimum values of the one-point specification of the Markov random field. The case in which there is phase transition is addressed in the frame of the ferromagnetic Ising model. We prove that the existence of infinite interaction neighborhoods depends on the phase.