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Biometrika 1999 86(4):785-801; doi:10.1093/biomet/86.4.785
© 1999 by Biometrika Trust
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Decomposable graphical Gaussian model determination

P GiudiciA1 and PJ GreenA2

A1 Department of Economics and Quantitative Methods, University of Pavia, Via San Felice n. 5, 27100 Pavia, Italy E-mail: pgiudici@eco.unipv.it A2 Department of Mathematics, University of Bristol, Bristol, BS8 1TW, UK E-mail: p.j.green@bristol.ac.uk

We propose a methodology for Bayesian model determination in decomposable graphical Gaussian models. To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. To ensure compatibility across models, such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. We explore alternative structures for the hyperparameters of the latter, and their consequences for the model. Model determination is carried out by implementing a reversible jump Markov chain Monte Carlo sampler. In particular, the dimension-changing move we propose involves adding or dropping an edge from the graph. We characterise the set of moves which preserve the decomposability of the graph, giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep. As state variable, we use the incomplete variance-covariance matrix, containing only the elements for which the corresponding element of the inverse is nonzero. This allows all computations to be performed locally, at the clique level, which is a clear advantage for the analysis of large and complex datasets. Finally, the statistical and computational performance of the procedure is illustrated by mean of both artificial and real datasets.

Key Words: Bayesian model selection; Hyper-Markov distribution; inverse Wishart distribution; junction tree; reversible jump Markov chain Monte Carlo


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