Bayesian estimation of multivariate Gaussian Markov random fields with constraint

This article concerns with conditionally formulated multivariate Gaussian Markov random fields (MGMRF) for modeling multivariate local dependencies with unknown dependence parameters subject to positivity constraint. In the context of Bayesian hierarchical modeling of lattice data in general and Bayesian disease mapping in particular, analytic and simulation studies provide new insights into various approaches to posterior estimation of dependence parameters under “hard” or “soft” positivity constraint, including the well‐known strictly diagonal dominance criterion and options of hierarchical priors. Hierarchical centering is examined as a means to gain computational efficiency in Bayesian estimation of multivariate generalized linear mixed effects models in the presence of spatial confounding and weakly identified model parameters. Simulated data on irregular or regular lattice, and three datasets from the multivariate and spatiotemporal disease mapping literature, are used for illustration. The present investigation also sheds light on the use of deviance information criterion for model comparison, choice, and interpretation in the context of posterior risk predictions judged by borrowing‐information and bias‐precision tradeoff. The article concludes with a summary discussion and directions of future work. Potential applications of MGMRF in spatial information fusion and image analysis are briefly mentioned.