Scalable Spike-and-Slab

Abstract

Spike-and-slab priors are commonly used for Bayesian variable selection, due to their interpretability and favorable statistical properties. However, existing samplers for spikeand-slab posteriors incur prohibitive computational costs when the number of variables is large. In this article, we propose Scalable Spike-and-Slab (S 3 ), a scalable Gibbs sampling implementation for high-dimensional Bayesian regression with the continuous spike-and-slab prior of George and McCulloch [1993]. For a dataset with n observations and p covariates, S 3 has order max{n 2 pt, np} computational cost at iteration t where pt never exceeds the number of covariates switching spike-and-slab states between iterations t and t − 1 of the Markov chain. This improves upon the order n 2 p per-iteration cost of state-of-the-art implementations as, typically, pt is substantially smaller than p. We apply S 3 on synthetic and real-world datasets, demonstrating orders of magnitude speed-ups over existing exact samplers and significant gains in inferential quality over approximate samplers with comparable cost.