Bayesian inference of multiple Ising models for heterogeneous public opinion survey networks

Public opinion studies show that relationships between opinions shift based on respondent characteristics. Understanding these complexities requires methods that account for heterogeneity across groups. We adopt a class of multiple Ising models that use graphs to analyse how external factors—such as time spent online or generational differences—shape joint dependence relationships between opinions. A Bayesian methodology is proposed based on a Markov Random Field prior, allowing information sharing across groups to encourage common edges when supported by data. A spike-and-slab prior induces sparsity and identifies shared graph structures across subgroups. Specifically, we develop two Bayesian approaches for inferring multiple Ising models, focusing on model selection: (i) a Fully Bayesian method for low dimensional graphs using conjugate priors and exact likelihood and (ii) an Approximate Bayesian method for high-dimensional graphs based on a quasi-likelihood approach, avoiding computational intractability. These methods are applied to two US public opinion studies: one examining how time spent online affects confidence in political institutions, and another exploring intergenerational differences in opinions on public spending. Our results balance identifying significant edges (both shared and group-specific) with sparsity while quantifying uncertainty, ultimately revealing how external factors shape public opinion dynamics.