Mixture of Directed Graphical Models for Discrete Spatial Random Fields

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). Notably, the MDGM is not proposed as an approximation to an MRF, but rather as an alternative that provides valid posterior inference while being computationally faster than exact MRF inference and more principled than the pseudo-likelihood approximation (aMRF) commonly used in practice. We introduce the concept of compatibility to show how an undirected graph can be used as a template for the dependencies between areal units to create sets of DAGs which, as a collection, preserve the dependencies represented in the template undirected graph. Lastly, we compare highlighted classes of 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.