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Wishart Distributions for Covariance Graph Models

Gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph G. These models are distinctly different from the traditional concentration graph models (often also referred to as Gaus-sian graphical models or covariance selection models), as the zeroes in the parameter are now reflected in the covariance matrix Σ, as compared to the concentration matrix Ω = Σ −1. The parameter space of interest for covariance graph models is the cone PG of positive definite matrices with fixed zeroes corresponding to the missing edges of G. As in Letac and Massam [20] we consider the case when G is decomposable. In this paper we construct on the cone PG a family of Wishart distributions, that serve a similar purpose in the covariance graph setting, as those constructed by Letac and Massam [20] and Dawid and Lauritzen [8] do in the concentration graph setting. We proceed to undertake a rigorous study of these " covariance " Wishart distributions, and derive several deep and useful properties of this class. First, they form a rich conjugate family of priors with multiple shape parameters for covariance graph models. Second, we show how to sample from these distributions by using a block Gibbs sampling algorithm, and prove convergence of this block Gibbs sampler. Development of this class of distributions enables Bayesian inference, which in turn allows for the estimation of Σ even in the case when the sample size is less than the dimension of the data (i.e., when 'n < p'), otherwise not possible in general in the maximum likelihood framework. Third, we show that when G is a homogeneous graph, the normalizing constant can be evaluated in closed form. In fact, in this particular case, the family IWQ G of [20] is a special case of our covariance Wishart distributions. Fourth, we also show that this family of priors satisfies an analogue of the strong directed hyper Markov property, when G is homogeneous, and proceed to compute moments and Laplace transforms. We then introduce a new concept called 'generalized posterior linearity' for curved exponential families. We proceed to demonstrate that a subclass of our priors satisfies this notion of posterior linearity-thereby drawing parallels with the Diaconis-Ylvisaker priors for natural exponential families. Fifth and finally, we illustrate the use of our family of conjugate priors on real and simulated data. 1. Introduction. Due to recent …