Bayesian Inference for Discrete Markov Random Fields Through Coordinate Rescaling

Abstract

Discrete Markov random fields are undirected graphical models that capture complex conditional dependencies between discrete variables. Conducting exact posterior inference in these models is often computationally challenging because evaluating their normalizing constant requires summation over all possible state configurations, and the size of this state space grows exponentially with the number of variables and their possible states. As a result, exact likelihood-based inference is infeasible in many practical settings, and existing methods, such as Double Metropolis-Hastings or pseudo-likelihood approximations, either scale poorly to large systems or underestimate posterior variability. To address these limitations, we propose a new class of coordinate-rescaling sampling methods that transform pseudo-likelihood-based posteriors to approximate the covariance structure of the full likelihood posterior while preserving computational efficiency. In a simulation study and an empirical application, the coordinate-rescaling samplers correct the underestimated posterior uncertainty of the pseudo-likelihood and reproduce the Bayes factors of a full-likelihood reference, with the accuracy of post hoc calibration of the pseudo-posterior and at the cost of the pseudo-likelihood sampler, providing a scalable and reliable approach to Bayesian inference in discrete MRFs.