Bayesian inference in hidden Markov random fields for binary data defined on large lattices

A hidden Markov random fields may arise where a Markov random field – a spatial arrangement of correlated discrete states – is corrupted by some observational noise process. We assume that the number of hidden or latent states is known and wish to perform inference on all unknown parameters. The main challenge in such cases is to calculate the likehood of the hidden states, which could be computationally very expensive. Recently new methods have been proposed to circumvent this problem, all of which are restricted to relatively small lattices. The main contribution of this paper is to introduce approximate methods to compute the likelihood for large lattices based on exact likelihood calculations for smaller lattices. We introduce approximate likelihood methods by relaxing some of the dependencies in the latent model, and also by approximating the likelihood by a partially ordered Markov model defined on a collection of sublattices. Results are presented based on simulated data as well as inference for the temporal-spatial structure of the interaction between upand down-regulated states within the mitochondrial chromosome of the Plasmodium falciparum organism.