Abstract: Variable screening is a powerful and efficient tool for dimension reduction under ultrahigh-dimensional settings. However, most existing methods overlook useful prior knowledge in specific applications. In this work, from a Bayesian modeling perspective, we develop a unified variable screening procedure for linear regression models. We discuss different constructions of posterior mean screening (PMS) statistics to incorporate different types of prior knowledge according to specific applications. With non-informative prior specifications, PMS is equivalent to the high-dimensional ordinary least-square projection (HOLP). We establish the screening consistency property for PMS with different types of prior knowledge. We show that PMS is robust to prior misspecifications. Furthermore, when the prior knowledge provides correct information on the true parameter settings, PMS can substantially improve the selection accuracy over that of the HOLP and other existing methods. We illustrate our method using extensive simulation studies and an analysis of neuroimaging data.

Key words and phrases: Linear regression, posterior mean screening, prior knowledge, screening consistency.