Bayesian high-dimensional linear regression with generic spike-and-slab priors
This provides a foundational theoretical framework for Bayesian high-dimensional regression, addressing a key bottleneck for statisticians and machine learning practitioners.
The paper tackles the lack of general theoretical guarantees for spike-and-slab priors in high-dimensional linear regression by proposing a class of generic priors and developing a unified framework to prove nearly-optimal posterior contraction rates and model selection consistency, including previous results as special cases.
Spike-and-slab priors are popular Bayesian solutions for high-dimensional linear regression problems. Previous theoretical studies on 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 the nearly-optimal posterior contraction rate and the model selection consistency. Our results include those of Narisetty and He (2014) and Castillo et al. (2015) as special cases.