A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models
1 Faculty of Economic and Administrative Sciences, Department of Business Administration, Suleyman Demirel University, 32260 Isparta, Turkey aliye{at}emk.com.tr, 2 Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, M3J 1P3, Canada massamh{at}yorku.ca
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. In this paper, 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.
Key Words: Estimation in covariance selection models; Exact sampling distribution Wishart; Marginal likelihood; Nondecomposable graphical Gaussian model; Normalising constant
Received August 2003. Revised June 2004.
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