Fast Bayesian variable selection for high dimensional linear models: Marginal solo spike and slab priors

Abstract

This paper presents a method for fast Bayesian variable selection in the normal linear regression model with high dimensional data. A novel approach is adopted in which an explicit posterior probability for including a covariate is obtained. The method is sequential but not order dependent, one deals with each covariate one by one, and a spike and slab prior is only assigned to the coefficient under investigation. We adopt the well-known spike and slab Gaussian priors with a sample size dependent variance, which achieves strong selection consistency for marginal posterior probabilities even when the number of covariates grows almost exponentially with sample size. Numerical illustrations are presented where it is shown that the new approach provides essentially equivalent results to the standard spike and slab priors, i.e. the same marginal posterior probabilities of the coefficients being nonzero, which are estimated via Gibbs sampling. Hence, we obtain the same results via the direct calculation of $p$ probabilities, compared to a stochastic search over a space of $2^{p}$ elements. Our procedure only requires $p$ probabilities to be calculated, which can be done exactly, hence parallel computation when $p$ is large is feasible.