Variational Bayes and the reduced dependence approximation for the autologistic model on an irregular grid with applications

Discrete Markov random field models provide a natural framework for representing images or spatial datasets. They model the spatial association present while providing a convenient Markovian dependency structure and strong edge-preservation properties. However, parameter estimation for discrete Markov random field models is difficult due to the complex form of the associated normalizing constant for the likelihood function. For large lattices, the reduced dependence approximation to the normalizing constant is based on the concept of performing computationally efficient and feasible forward recursions on smaller sublattices which are then suitably combined to estimate the constant for the whole lattice. We present an efficient computational extension of the forward recursion approach for the autologistic model to lattices that have an irregularly shaped boundary and which may contain regions with no data; these lattices are typical in applications. Consequently, we also extend the reduced dependence approximation to these scenarios enabling us to implement a practical and efficient non-simulation based approach for spatial data analysis within the variational Bayesian framework. The methodology is illustrated through application to simulated data and example images. The supplemental materials include our C++ source code for computing the approximate normalizing constant and simulation studies.