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In this paper, we propose a class of Bayes estimators for the covariancematrix of graphical Gaussian models Markov with respect to a decomposable graph$G$. Working with the $W_{P_G}$ family defined by Letac and Massam [Ann.Statist. 35 (2007) 1278--1323] we derive closed-form expressions for Bayesestimators under the entropy and squared-error losses. The $W_{P_G}$ familyincludes the classical inverse of the hyper inverse Wishart but has many moreshape parameters, thus allowing for flexibility in differentially shrinkingvarious parts of the covariance matrix. Moreover, using this family avoidsrecourse to MCMC, often infeasible in high-dimensional problems. We illustratethe performance of our estimators through a collection of numerical exampleswhere we explore frequentist risk properties and the efficacy of graphs in theestimation of high-dimensional covariance structures.

Flexible covariance estimation in graphical Gaussian models