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A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models. (English) Zbl 1094.62028
Summary: A centred Gaussian model that is Markov with respect to an undirected graph \(G\) is characterised by the parameter set of its precision matrices which is the cone \(M^+(G)\) of positive definite matrices with entries corresponding to the missing edges of \(G\) constrained to be equal to zero. In a Bayesian framework, the conjugate family for the precision parameter is the distribution with Wishart density with respect to the Lebesgue measure restricted to \(M^+(G)\). We call this distribution the \(G\)-Wishart. When \(G\) is nondecomposable, the normalising constant of the \(G\)-Wishart cannot be computed in closed form.
We give a simple Monte Carlo method for computing this normalising constant. The main feature of our method is that the sampling distribution is exact and consists of a product of independent univariate standard normal and chi-squared distributions that can be read off the graph \(G\). Computing this normalising constant is necessary for obtaining the posterior distribution of \(G\) or the marginal likelihood of the corresponding graphical Gaussian model. Our method also gives a way of sampling from the posterior distribution of the precision matrix.

62F15 Bayesian inference
65C05 Monte Carlo methods
05C90 Applications of graph theory
62H12 Estimation in multivariate analysis
65C60 Computational problems in statistics (MSC2010)
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