We propose a Bayesian variable selection procedure for ultrahigh-dimensional
linear regression models. The number of regressors involved in regression,
pnβ, is allowed to grow exponentially with n. Assuming the true model to be
sparse, in the sense that only a small number of regressors contribute to this
model, we propose a set of priors suitable for this regime. The model selection
procedure based on the proposed set of priors is shown to be variable selection
consistent when all the 2pnβ models are considered. In the
ultrahigh-dimensional setting, selection of the true model among all the
2pnβ possible ones involves prohibitive computation. To cope with this, we
present a two-step model selection algorithm based on screening and Gibbs
sampling. The first step of screening discards a large set of unimportant
covariates, and retains a smaller set containing all the active covariates with
probability tending to one. In the next step, we search for the best model
among the covariates obtained in the screening step. This procedure is
computationally quite fast, simple and intuitive. We demonstrate competitive
performance of the proposed algorithm for a variety of simulated and real data
sets when compared with several frequentist, as well as Bayesian methods