Bayesian Variable Selection Under High-Dimensional Settings with Grouped Covariates

Consider the normal linear regression setup when the number of covariates is much larger than the sample size, and the covariates form correlated groups. The response variable is not related to an entire group of covariates in all or none basis, rather sparsity persists irrespective of the group structure. We extend the traditional g-prior setup to this framework, under both scenarios when the group structure is known and when it is unknown. Variable selection consistency of the maximum a-posteriori approach with the proposed priors is shown to hold under fairly general conditions. For implementation, we propose a method that efficiently uses the group structure, if known or estimated, to expedite the search algorithm for the optimal model. Towards that, first, a group importance probability termed as group SIS (GSIS) is proposed, and then a novel stochastic search variable selection algorithm termed as group informed variable selection algorithm (GiVSA) is introduced. GiVSA utilizes group structure and importance probability to explore the model space efficiently without initial covariate screening. Screening consistency of GSIS is analyzed, and a probabilistic upper bound of the mixing time of GiVSA is provided using the canonical path ensemble approach of Yang et al. (2016). The performance of the proposed priors (with an application of GiVSA) and GSIS is compared with several state-of-the-art frequestist and Bayesian competitors using ample simulated and real data sets.