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. 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 swings in market volatility, offering improved calibration of predictive distributions.
Highlights
The Gaussian graphical model (GGM) has received widespread consideration and estimators obeying graphical constraints in standard Gaussian sampling were proposed as early as Dempster (1972)
By using the conditional Bayes factor (CBF) in (6), Wang and Li (2012) propose model moves that do not rely on RJ methods, and after assessing which graph to move to, the parameter Kjj, as well as Kij if e is in the accepted graph, are resampled according to their conditional distributions given K−e
Suppose that (K, G) is the current state of the Markov chain Monte Carlo (MCMC) chain and we propose to move to G′ by adding the edge e to G
Summary
The Gaussian graphical model (GGM) has received widespread consideration (see Jones et al, 2005) and estimators obeying graphical constraints in standard Gaussian sampling were proposed as early as Dempster (1972). Wang and Li (2012) explore the use of a double MH algorithm (Liang, 2010) to avoid the computationally expensive and numerically unstable MC approximation of normalizing constants proposed by Atay-Kayis and Massam (2005) We continue this departure from RJ and investigate an alternative method for simultaneously updating the GGM and associated K in hierarchical Bayesian settings. Our proposal differs from the dynamic linear model context in that we model an exponential term that represents market volatility This latent Gaussian factor requires a strategy for hierarchically estimating the GGM and associated precision term, which are used to subsequently update the volatility process.
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