Abstract: We consider the joint sparse estimation of the regression coefficients and the covariance matrix for covariates in a high-dimensional regression model. Here, the predictors are both relevant to a response variable of interest and functionally related to one another via a Gaussian directed acyclic graph (DAG) model. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional inde- pendence assumptions on the underlying predictors. A variety of methods have been developed in recent years for Bayesian inferences that identify such network- structured predictors in a regression setting. However, crucial sparsity selection properties for these models have not been thoroughly investigated. Therefore, we consider a hierarchical model with spike and slab priors on the regression coefficients, and a flexible and general class of DAG–Wishart distributions with multiple shape parameters on the Cholesky factors of the inverse covariance matrix. Under mild regularity assumptions, we establish the joint selection consistency for both the variable and the underlying DAG of the covariates when the dimension of the predictors is allowed to grow much larger than the sample size. We demonstrate that our method outperforms existing methods in selecting network-structured predictors in several simulation settings.

Key words and phrases: DAG–Wishart prior, posterior ratio consistency, strong selection consistency.