Bayesian Inference for Discrete Markov Random Fields Through Coordinate Rescaling

Abstract: Discrete Markov random fields (MRFs) represent a class of undirected graphical models that capture complex conditional dependencies between discrete variables. Conducting exact posterior inference in these models is computationally challenging due to the intractable partition function, which depends on the model parameters and sums over all possible state configurations in the system. As a result, using the exact likelihood function is infeasible and existing methods, such as Double Metropolis-Hastings or pseudo-likelihood approximations, either scale poorly to large systems or underestimate the variability of the target posterior distribution. To address both computational burden and efficiency loss, we propose a new class of coordinate-rescaling sampling methods, which map the model parameters from the pseudo-likelihood space to the target posterior, preserving computational efficiency while improving posterior inference. Finally, in simulation studies, we compare the proposed method to existing approaches and illustrate that coordinate-rescaling sampling provides more accurate estimates of posterior variability, offering a scalable and robust solution for Bayesian inference in discrete MRFs.