  Covariance estimation and selection for high-dimensional multivariate
datasets is a fundamental problem in modern statistics. Gaussian directed
acyclic graph (DAG) models are a popular class of models used for this purpose.
Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse
covariance matrix, and the sparsity pattern in turn corresponds to specific
conditional independence assumptions on the underlying variables. A variety of
priors have been developed in recent years for Bayesian inference in DAG
models, yet crucial convergence and sparsity selection properties for these
models have not been thoroughly investigated. Most of these priors are
adaptations or generalizations of the Wishart distribution in the DAG context.
In this paper, we consider a flexible and general class of these 'DAG-Wishart'
priors with multiple shape parameters. Under mild regularity assumptions, we
establish strong graph selection consistency and establish posterior
convergence rates for estimation when the number of variables p is allowed to
grow at an appropriate sub-exponential rate with the sample size n.
