The Isserlis matrix and its application to non-decomposable graphical Gaussian models

The operations of matrix completion and Isserlis matrix construction arise in the statistical analysis of graphical Gaussian models, in both the Bayesian and frequentist approaches. In this paper we study the properties of the Isserlis matrix of the completion, , of a positive definite matrix and propose an edge set indexing notation which highlights the symmetry existing between and its Isserlis matrix. In this way well-known properties of can be exploited to give an easy proof of certain asymptotic results for decomposable graphical Gaussian models, as well as to extend such results to the non-decomposable case. In particular we consider the asymptotic variance of maximum likelihood estimators, the non-informative Jeffreys prior, the Laplace approximation to the Bayes factor, the asymptotic distribution of the maximum likelihood estimators and the asymptotic posterior distribution of the parameters in a Bayesian conjugate analysis. In dealing with distributions over non-decomposable models an extension of the hyper-Markov property to models with non-chordal graphs is required. Our proposed extension is justified by an analysis of the conditional independence structure of the sufficient statistic.

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