Mixture of Directed Graphical Models for Discrete Spatial Random Fields

TL;DR

This paper introduces a novel mixture of directed graphical models (MDGMs) for discrete spatial data, offering computational efficiency and better approximation of spatial dependence structures compared to traditional Markov random fields.

Contribution

The paper proposes MDGMs as a new framework that aligns with undirected spatial dependence structures and improves posterior inference accuracy when the data-generating process is an MRF.

Findings

MDGMs provide computationally efficient evaluation.

MDGMs better approximate MRF posteriors when the data is generated by an MRF.

Simulation and real data analyses demonstrate advantages over traditional MRF models.

Abstract

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 shares the same default, underlying graphical representation of spatial dependence as an MRF. However, in the case that the data generating mechanism of the latent spatial field is known to be an MRF, we find that posterior inferences under an MDGM prior better approximate the posterior of the model with a correctly specified MRF prior. 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.