Bayesian Inference of Multiple Ising Models for Heterogeneous Public Opinion Survey Networks

In public opinion studies, the relationships between opinions on different topics are likely to shift based on the characteristics of the respondents. Thus, understanding the complexities of public opinion requires methods that can account for the heterogeneity in responses across different groups. Multiple graphs are used to study how external factors-such as time spent online or generational differences-shape the joint dependence relationships between opinions on various topics. Specifically, we propose a class of multiple Ising models where a set of graphs across different groups are able to capture these variations and to model the heterogeneity induced in a set of binary variables by external factors. The proposed Bayesian methodology is based on a Markov Random Field prior for 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 spike-and-slab 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 the inference of multiple Ising models with a special focus on model selection: (i) a Fully Bayesian method for low-dimensional graphs based on conjugate priors specified with respect to the exact likelihood, and (ii) 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. These methods are employed for the analysis of data from two public opinion studies in US. The obtained results display a good trade-off between identifying significant edges (both shared and group-specific) and having sparse networks, all while quantifying the uncertainty of the graph structure and the graphs' similarity.