Approximations and bounds for binary Markov random fields

TL;DR

This paper develops approximation methods and bounds for the intractable normalising constant in binary Markov random fields, facilitating parameter estimation and Bayesian analysis in spatial data models.

Contribution

It introduces novel approximation techniques for the normalising constant in binary Markov random fields using pseudo-Boolean functions, enabling practical Bayesian inference.

Findings

The approximations are accurate for Ising and higher-order models.

Bounds improve understanding of the normalising constant's range.

Applications demonstrate effective parameter estimation and Bayesian modeling.

Abstract

Discrete Markov random fields form a natural class of models to represent images and spatial data sets. The use of such models is, however, hampered by a computationally intractable normalising constant. This makes parameter estimation and a fully Bayesian treatment of discrete Markov random fields difficult. We apply approximation theory for pseudo-Boolean functions to binary Markov random fields and construct approximations and upper and lower bounds for the associated computationally intractable normalising constant. As a by-product of this process we also get a partially ordered Markov model approximation of the binary Markov random field. We present numerical examples with both the pairwise interaction Ising model and with higher-order interaction models, showing the quality of our approximations and bounds. We also present simulation examples and one real data example demonstrating how the approximations and bounds can be applied for parameter estimation and to handle a fully-Bayesian model computationally.