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的面状单元离散值结果建模的方法存在计算和理论上的挑战，特别是在贝叶斯框架下需要完整后验推断时。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）。该混合模型的组成单元——有向图模型——可通过有向无环图（DAGs）表示，且计算评估速度较快。这些代表混合分量的DAGs经过选择，使其对应于面状单元空间邻接/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））规范的基础。我们引入相容性概念，以说明如何利用无向图作为面状单元间结构依赖关系的模板，从而生成一系列DAGs集合，这些集合整体上保留了模板无向图所表征的结构依赖关系。随后，我们提出三类相容DAGs及相应的MDGMs拟合算法。此外，我们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分析，将MDGMs与MRFs以及高维环境中常用的贝叶斯MRF模型近似方法进行了比较。