ABSTRACT
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.
