Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models


Atay-Kayis A. , MASSAM H.

BIOMETRIKA, cilt.92, ss.317-335, 2005 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 92 Konu: 2
  • Basım Tarihi: 2005
  • Doi Numarası: 10.1093/biomet/92.2.317
  • Dergi Adı: BIOMETRIKA
  • Sayfa Sayısı: ss.317-335

Özet

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.