  Estimation of the covariance matrix for high-dimensional multivariate
datasets is a challenging and important problem in modern statistics. In this
paper, we focus on high-dimensional Gaussian DAG models where sparsity is
induced on the Cholesky factor L of the inverse covariance matrix. In recent
work, ([Cao, Khare, and Ghosh, 2019]), we established high-dimensional sparsity
selection consistency for a hierarchical Bayesian DAG model, where an
Erdos-Renyi prior is placed on the sparsity pattern in the Cholesky factor L,
and a DAG-Wishart prior is placed on the resulting non-zero Cholesky entries.
In this paper we significantly improve and extend this work, by (a) considering
more diverse and effective priors on the sparsity pattern in L, namely the
beta-mixture prior and the multiplicative prior, and (b) establishing sparsity
selection consistency under significantly relaxed conditions on p, and the
sparsity pattern of the true model. We demonstrate the validity of our
theoretical results via numerical simulations, and also use further simulations
to demonstrate that our sparsity selection approach is competitive with
existing state-of-the-art methods including both frequentist and Bayesian
approaches in various settings.
