ABSTRACT
Continuous spike-and-slab priors are commonly used for high dimensional variable selection and shrinkage in the Bayesian framework. Gaussian and Laplace spike-and-slab priors are two of the most popular among such priors which have been widely used in a variety of settings. This chapter will provide an overview of some of the recent theoretical advances for the posteriors based on these priors in terms of both estimation accuracy and model selection consistency. Moreover, theoretical insights on the variable selection consistency using a general class of priors that include the Gaussian and Laplace ones will be provided. These theoretical results provide specific conditions on the spike-and-slab prior distributions with a general base density for achieving variable selection consistency. In particular, the requirements on the spike-and-slab prior distributions are characterized by their relative magnitudes at the origin and at the tails. Recent advances on scalable computational approaches for posterior computation will be discussed. This chapter will conclude with a discussion on some applications of continuous spike-and-slab priors beyond the linear regression models.
