Variable Selection with Shrinkage Priors via Sparse Posterior Summaries

Shrinkage priors with the ability to perform both global and local shrinkage have become popular techniques for Bayesian regularization. These priors can typically be written as scale-mixtures of Gaussians, which enable the construction of posterior sampling schemes. While these priors are well-suited for estimation and parameter inference, they do not include variable selection as a by-product when posterior summaries such as means and covariances are extracted. Bondell and Reich (2012) proposed an approach to post-process the posterior distribution of a parameter vector based on selecting the sparsest model among those contained within a given posterior credible region. By changing the coverage level of the credible region, they constructed a sequence of models to accommodate variable selection. In this chapter, we discuss the use of global-local shrinkage priors and combine them with variable selection using sparse posterior summaries in some common models.