Variational Bayes algorithm and posterior consistency of Ising model parameter estimation

TL;DR

This paper develops a variational Bayes method for efficient Bayesian inference in Ising models, addressing the intractability of the likelihood and providing theoretical guarantees and validation through simulations.

Contribution

It introduces a variational Bayes algorithm for Ising model parameter estimation using pseudo-likelihood, with proven posterior contraction rates and extensive simulation validation.

Findings

The variational approach achieves accurate parameter estimation.

Posterior contraction rates are established under the Gaussian mean-field family.

Simulation results confirm the method's efficacy and robustness.

Abstract

Ising models originated in statistical physics and are widely used in modeling spatial data and computer vision problems. However, statistical inference of this model remains challenging due to intractable nature of the normalizing constant in the likelihood. Here, we use a pseudo-likelihood instead to study the Bayesian estimation of two-parameter, inverse temperature, and magnetization, Ising model with a fully specified coupling matrix. We develop a computationally efficient variational Bayes procedure for model estimation. Under the Gaussian mean-field variational family, we derive posterior contraction rates of the variational posterior obtained under the pseudo-likelihood. We also discuss the loss incurred due to variational posterior over true posterior for the pseudo-likelihood approach. Extensive simulation studies validate the efficacy of mean-field Gaussian and bivariate Gaussian families as the possible choices of the variational family for inference of Ising model parameters.