Bayesian high-dimensional linear regression with generic spike-and-slab priors

Spike-and-slab priors are popular Bayesian solutions for high-dimensional linear regression problems. Previous works on theoretical properties of spike-and-slab methods focus on specific prior formulations and use prior-dependent conditions and analyses, and thus can not be generalized directly. In this paper, we propose a class of generic spike-and-slab priors and develop a unified framework to rigorously assess their theoretical properties. Technically, we provide general conditions under which generic spike-and-slab priors can achieve a nearly-optimal posterior contraction rate and model selection consistency. Our results include those of Castillo et al. (2015) and Narisetty and He (2014) as special cases.