Xuan Cao, Ph.D., Department of Mathematical Sciences, University of Cincinnati

"Consistent Bayesian Joint Variable and DAG Selection in High Dimensions"

Motivated by the eQTL analysis, we consider joint sparse estimation of the regression coefficient matrix and the error covariance matrix in a high-dimensional multivariate regression model for studying conditional independence relationships among a set of genes and discovering possible genetic effects. The error covariance matrix is modeled via Gaussian directed acyclic graph (DAG) and sparsity is introduced in the Cholesky factor of the inverse covariance matrix, while the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying variables. In this talk, we consider a flexible and general class of these ‘DAG-Wishart’ priors with multiple shape parameters on the space of Cholesky factors and a spike and slab prior on the regression coefficients. Under mild regularity assumptions, we establish the joint selection consistency for both the variable and the underlying DAG when both the number of predictors and the dimension of the covariance matrix are allowed to grow much larger than the sample size. We demonstrate our theoretical results through a marginalization-based collapsed Gibbs sampler that offers a computationally feasible and efficient solution for exploring the sample space.

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