Covariance estimation in high-dimensional settings has recently received widespread attention. In this context, graphical models (where dependencies between variables are represented by means of a graph) can act as a tool for regularization and have proven to be useful for the analysis of high dimensional data. A subclass of graphical models, known as Gaussian covariance graph models, encode marginal independence among random variables by means of a graph G. These are distinctly different from the traditional concentration graph models (often referred to as covariance selection models). Inference for these models is challenging both in the frequentist and Bayesian frameworks, since the models give rise to a curved exponential family. Maximum likelihood estimation for these models has received much attention recently. In this talk, we address the issue of Bayesian inference for these models. Since we are now in a curved setting, the Diaconis-Ylvisaker theory is no longer applicable; hence the standard Wishart distributions or those proposed for concentration graph models are not useful for analyzing covariance graph models. We propose a rich family of Wishart distributions which act as a conjugate family of priors for our class of models and investigate the theoretical properties of this class of distributions. We proceed to study the benefits of using this family for Bayesian inference in high-dimensional settings.
(This is joint work with Kshitij Khare)