Wishart distributions for decomposable graphs

Abstract

When considering a graphical Gaussian model ${\mathcal{N}}_{G}$ Markov with respect to a decomposable graph G, the parameter space of interest for the precision parameter is the cone P_G of positive definite matrices with fixed zeros corresponding to the missing edges of G. The parameter space for the scale parameter of ${\mathcal{N}}_{G}$ is the cone Q_G, dual to P_G, of incomplete matrices with submatrices corresponding to the cliques of G being positive definite. In this paper we construct on the cones Q_G and P_G two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272–1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior.

Both Type I and II Wishart distributions depend on multivariate shape parameters. A shape parameter is acceptable if and only if it satisfies a certain eigenvalue property. We show that the sets of acceptable shape parameters for a noncomplete G have dimension equal to at least one plus the number of cliques in G. These families, as conjugate families, are richer than the traditional Diaconis–Ylvisaker conjugate families which all have a shape parameter set of dimension one. A decomposable graph which does not contain a three-link chain as an induced subgraph is said to be homogeneous. In this case, our Wisharts are particular cases of the Wisharts on homogeneous cones as defined by Andersson and Wojnar [J. Theoret. Probab. 17 (2004) 781–818] and the dimension of the shape parameter set is even larger than in the nonhomogeneous case: it is indeed equal to the number of cliques plus the number of distinct minimal separators. Using the model where G is a three-link chain, we show by computing a 7-tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one.