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Marginal likelihood, also known as model evidence, is a fundamental quantityin Bayesian statistics. It is used for model selection using Bayes factors orfor empirical Bayes tuning of prior hyper-parameters. Yet, the calculation ofevidence has remained a longstanding open problem in Gaussian graphical models.Currently, the only feasible solutions that exist are for special cases such asthe Wishart or G-Wishart, in moderate dimensions. We develop an approach basedon a novel telescoping block decomposition of the precision matrix that allowsthe estimation of evidence by application of Chib's technique under a verybroad class of priors under mild requirements. Specifically, the requirementsare: (a) the priors on the diagonal terms on the precision matrix can bewritten as gamma or scale mixtures of gamma random variables and (b) those onthe off-diagonal terms can be represented as normal or scale mixtures ofnormal. This includes structured priors such as the Wishart or G-Wishart, andmore recently introduced element-wise priors, such as the Bayesian graphicallasso and the graphical horseshoe. Among these, the true marginal is known inan analytically closed form for Wishart, providing a useful validation of ourapproach. For the general setting of the other three, and several more priorssatisfying conditions (a) and (b) above, the calculation of evidence hasremained an open question that this article resolves under a unifyingframework.

Evidence Estimation in Gaussian Graphical Models Using a Telescoping Block Decomposition of the Precision Matrix