Current approaches for modeling discrete-valued outcomes associated with spatially-dependent areal units incur computational and theoretical challenges, especially in the Bayesian setting when full posterior inference is desired. As an alternative, we propose a novel statistical modeling framework for this data setting, namely a mixture of directed graphical models (MDGMs). The components of the mixture, directed graphical models, can be represented by directed acyclic graphs (DAGs) and are computationally quick to evaluate. The DAGs representing the mixture components are selected to correspond to an undirected graphical representation of an assumed spatial contiguity/dependence structure of the areal units, which underlies the specification of traditional modeling approaches for discrete spatial processes such as Markov random fields (MRFs). We introduce the concept of compatibility to show how an undirected graph can be used as a template for the structural dependencies between areal units to create sets of DAGs which, as a collection, preserve the structural dependencies represented in the template undirected graph. We then introduce three classes of compatible DAGs and corresponding algorithms for fitting MDGMs based on these classes. In addition, we compare MDGMs to MRFs and a popular Bayesian MRF model approximation used in high-dimensional settings in a series of simulations and an analysis of ecometrics data collected as part of the Adolescent Health and Development in Context Study.

翻译：当前针对与空间相关区域单元相关联的离散值结果建模方法存在计算和理论上的挑战，在贝叶斯框架下进行全后验推断时尤为明显。作为替代方案，我们针对此类数据提出了一种新颖的统计建模框架——混合有向图模型（MDGMs）。该混合模型的构成单元——有向图模型——可通过有向无环图（DAGs）表示，且计算评估效率高。这些代表混合分量的DAGs经过特别选择，以对应区域单元空间邻接/依赖结构的无向图表示，该结构正是传统离散空间过程建模方法（如马尔可夫随机场（MRFs））的基础。我们引入相容性概念，阐明如何将无向图作为区域单元间结构依赖关系的模板，从而构建能完整保持模板无向图所表征结构依赖关系的DAG集合。随后提出三类相容DAG及其对应的MDGM拟合算法。通过系列模拟实验和对"青少年健康与发展情境研究"中收集的生态计量数据的分析，我们将MDGMs与MRFs以及高维场景中常用的贝叶斯MRF模型近似方法进行了比较。