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We propose Bayesian methods for Gaussian graphical models that lead to sparseand adaptively shrunk estimators of the precision (inverse covariance) matrix.Our methods are based on lasso-type regularization priors leading toparsimonious parameterization of the precision matrix, which is essential inseveral applications involving learning relationships among the variables. Inthis context, we introduce a novel type of selection prior that develops asparse structure on the precision matrix by making most of the elements exactlyzero, in addition to ensuring positive definiteness -- thus conducting modelselection and estimation simultaneously. We extend these methods to finite andinfinite mixtures of Gaussian graphical models for clustered data usingDirichlet process priors. We discuss appropriate posterior simulation schemesto implement posterior inference in the proposed models, including theevaluation of normalizing constants that are functions of parameters ofinterest which result from the restrictions on the correlation matrix. Weevaluate the operating characteristics of our method via several simulationsand in application to real data sets.

Bayesian sparse graphical models and their mixtures using lasso selection priors