Markov Random Fields with Higher‐order Interactions

Abstract

Discrete‐state Markov random fields on regular arrays have played a significant role in spatial statistics and image analysis. For example, they are used to represent objects against background in computer vision and pixel‐based classification of a region into different crop types in remote sensing. Convenience has generally favoured formulations that involve only pairwise interactions. Such models are in themselves unrealistic and, although they often perform surprisingly well in tasks such as the restoration of degraded images, they are unsatisfactory for many other purposes. In this paper, we consider particular forms of Markov random fields that involve higher‐order interactions and therefore are better able to represent the large‐scale properties of typical spatial scenes. Interpretations of the parameters are given and realizations from a variety of models are produced via Markov chain Monte Carlo. Potential applications are illustrated in two examples. The first concerns Bayesian image analysis and confirms that pairwise‐interaction priors may perform very poorly for image functionals such as number of objects, even when restoration apparently works well. The second example describes a model for a geological dataset and obtains maximum‐likelihood parameter estimates using Markov chain Monte Carlo. Despite the complexity of the formulation, realizations of the estimated model suggest that the representation is quite realistic.