  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, however, 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. Theoretical developments and
computational timing experiments reveal an algorithm that has limited
computational demands and dramatically improves on computing times of existing
methods. We conclude by developing a parsimonious multivariate stochastic
volatility model that embeds GGM uncertainty in a larger hierarchical
framework. The method is shown to be capable of adapting to the extreme swings
in market volatility experienced in 2008 after the collapse of Lehman Brothers,
offering considerable improvement in posterior predictive distribution
calibration.
