Bayesian inference of multiple Ising models for heterogeneous data

Abstract

Multiple Ising models can be used to model the heterogeneity induced in a set of binary variables by external factors. These factors may influence the joint dependence relationships represented by a set of graphs across different groups. The inference for this class of models is presented, and a Bayesian methodology is proposed based on a Markov Random Field prior to the multiple graph setting. Such prior enables the borrowing of strength across the different groups to encourage common edges when supported by the data. Sparse-inducing priors are employed on the parameters that measure graph similarities to learn which subgroups have a shared graph structure. Two Bayesian approaches are developed for inference and model selection, 1) a Fully Bayesian method for low-dimensional graphs based on conjugate priors specified w.r.t. the exact likelihood, and 2) an Approximate Bayesian method based on a quasi-likelihood approach for high-dimensional graphs where the normalization constant required in the exact method is computationally intractable. The methods’ performance is studied and compared with competing approaches through an extensive simulation study. Both inferential strategies are employed to analyze data resulting from two public opinion studies in the US. The first analyzes the confidence in political institutions in different groups divided by the time users spent on web pages. The second studies the opinion on public spending in diverse inter-generational groups.