Abstract
The Gaussian Graphical Model (GGM) is a popular tool for incorporating sparsity into joint multivariate distributions. The G-Wishart distribution, a conjugate prior for precision matrices satisfying general GGM constraints, has now been in existence for over a decade. However, due to the lack of a direct sampler, its use has been limited
in hierarchical Bayesian contexts, relegating mixing over the class of GGMs mostly to situations involving standard Gaussian likelihoods. Recent work has developed methods that couple model and parameter moves, first through reversible jump methods and later by direct evaluation of conditional Bayes factors and subsequent resampling. Further, methods for avoiding prior normalizing constant calculations–a serious bottleneck and source of numerical instability–have been proposed. We review and clarify these developments and then propose a new methodology for GGM comparison that blends many recent themes. We conclude with a series of examples showing how GGMs may be embedded in larger systems, including spatially dependent random effects, multivariate stochastic volatility models, and semiparametric Gaussian copulas.