Colloquium

Statistical models for spatially-dependent, discrete-valued observations associated with areal units (i.e., discrete space) are used routinely in areas of application ranging from disease mapping to small area estimation to image analysis. Despite a rich literature on both classical and Bayesian models for this setting, theoretical and computational considerations remain, particularly in the Bayesian setting when complete posterior inference is desired. To address the computational and theoretical concerns of existing models, we propose a novel modeling framework based on 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 for the areal units which underlies the specification of traditional modeling approaches for discrete spatial processes such as Markov random fields (MRFs). We introduce the concepts of weak and strong 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 and conditional independences 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. This presentation is based on joint work with Brandon Carter.