  We propose Bayesian methods for Gaussian graphical models that lead to sparse
and adaptively shrunk estimators of the precision (inverse covariance) matrix.
Our methods are based on lasso-type regularization priors leading to
parsimonious parameterization of the precision matrix, which is essential in
several applications involving learning relationships among the variables. In
this context, we introduce a novel type of selection prior that develops a
sparse structure on the precision matrix by making most of the elements exactly
zero, in addition to ensuring positive definiteness -- thus conducting model
selection and estimation simultaneously. We extend these methods to finite and
infinite mixtures of Gaussian graphical models for clustered data using
Dirichlet process priors. We discuss appropriate posterior simulation schemes
to implement posterior inference in the proposed models, including the
evaluation of normalizing constants that are functions of parameters of
interest which result from the restrictions on the correlation matrix. We
evaluate the operating characteristics of our method via several simulations
and in application to real data sets.
