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.

翻译：在民意研究中，不同议题观点间的关联性往往会因受访者特征而异。因此，理解民意的复杂性需要能够解释不同群体间响应异质性的方法。本研究采用多重图模型来探究外部因素（如在线时长或代际差异）如何影响不同议题观点间的联合依赖关系。具体而言，我们提出了一类多伊辛模型，其中跨不同群体的图集合能够捕捉这些变异，并对由外部因素引致的二元变量集合中的异质性进行建模。所提出的贝叶斯方法基于多重图设置下的马尔可夫随机场先验。该先验能够实现跨群体信息共享，在数据支持的情况下鼓励共同边的出现。我们在衡量图相似性的参数上采用稀疏诱导的尖峰-平板先验，以识别哪些子群体具有共享的图结构。针对多伊辛模型推断，我们开发了两种以模型选择为核心的贝叶斯方法：（一）基于精确似然共轭先验的完全贝叶斯方法，适用于低维图；（二）基于拟似然近似的近似贝叶斯方法，适用于精确方法中归一化常数计算不可行的高维图场景。我们将这些方法应用于两项美国民意调查数据的分析。所得结果在识别显著边（包括共享边与群体特定边）与保持网络稀疏性之间取得了良好平衡，同时量化了图结构及图相似性的不确定性。