Hyper inverse Wishart distribution for non-decomposable graphs and its application to Bayesian inference for Gaussian graphical models

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non-decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY-conjugate prior (Diacouls &Ylvisaker, 1979) for non-decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution (Dawid &Lauritzen, 1993). In particular, if G is an incomplete prime graph it constitutes a non-trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non-decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non-decomposable zero-pattern we introduce the operation of tr...