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Estimation of the covariance matrix for high-dimensional multivariatedatasets is a challenging and important problem in modern statistics. In thispaper, we focus on high-dimensional Gaussian DAG models where sparsity isinduced on the Cholesky factor L of the inverse covariance matrix. In recentwork, ([Cao, Khare, and Ghosh, 2019]), we established high-dimensional sparsityselection consistency for a hierarchical Bayesian DAG model, where anErdos-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) consideringmore diverse and effective priors on the sparsity pattern in L, namely thebeta-mixture prior and the multiplicative prior, and (b) establishing sparsityselection consistency under significantly relaxed conditions on p, and thesparsity pattern of the true model. We demonstrate the validity of ourtheoretical results via numerical simulations, and also use further simulationsto demonstrate that our sparsity selection approach is competitive withexisting state-of-the-art methods including both frequentist and Bayesianapproaches in various settings.

Consistent Bayesian Sparsity Selection for High-dimensional Gaussian DAG Models with Multiplicative and Beta-mixture Priors