Wishart distributions for decomposable covariance graph models

Abstract

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 Gaussian graphical models or covariance selection models) since the zeros 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 P_G of positive definite matrices with fixed zeros corresponding to the missing edges of G. As in Letac and Massam [Ann. Statist. 35 (2007) 1278–1323], we consider the case where G is decomposable. In this paper, we construct on the cone P_G a family of Wishart distributions which serve a similar purpose in the covariance graph setting as those constructed by Letac and Massam [Ann. Statist. 35 (2007) 1278–1323] and Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272–1317] 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 generally possible in the maximum likelihood framework. Third, we prove that when G is a homogeneous graph, our covariance priors correspond to standard conjugate priors for appropriate directed acyclic graph (DAG) models. This correspondence enables closed form expressions for normalizing constants and expected values, and also establishes hyper-Markov properties for our class of priors. We also note that when G is homogeneous, the family IW_QG of Letac and Massam [Ann. Statist. 35 (2007) 1278–1323] is a special case of our covariance Wishart distributions. Fourth, and finally, we illustrate the use of our family of conjugate priors on real and simulated data.