Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction

We study the classic problem of choosing a prior distribution for a location parameter β = (β 1,..., β p) as p grows large. First, we study the standard "global-local shrinkage" approach, based on scale mixtures of normals. Two theorems are presented which characterize certain desirable properties of shrinkage priors for sparse problems. Next, we review some recent results showing how Lévy processes can be used to generate infinite-dimensional versions of standard normal scale-mixture priors, along with new priors that have yet to be seriously studied in the literature. This approach provides an intuitive framework both for generating new regularization penalties and shrinkage rules, and for performing asymptotic analysis on existing models.