Joint Bayesian Variable and DAG Selection Consistency for High-dimensional Regression Models with Network-structured Covariates

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