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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.

##### MSC:
 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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