Bayesian Global-Local Shrinkage in High Dimensions

Purdue University, West Lafayette IN

Investigators

Abstract

High-dimensional data are ubiquitous in many modern applications such as genomics, finance, and image analysis. Developing approaches that are computationally scalable to the size of these data sets while retaining strong theoretical justifications remains a challenge. The goal of the project is to develop new methodology to enable tractable analysis of modern high-dimensional data sets. Software developed from this research will be made publicly available. Bayesian methodology for high-dimensional data traditionally relies on point mass mixture priors that have attractive theoretical properties but often scale poorly due to the computational difficulties associated with searching a high-dimensional discrete space. The goal of the project is to explore the use of global-local alternatives to high-dimensional problems. Many recent investigations using global-local priors, while showing signs of promise, have been restricted to studying the simple normal means model. The PIs will employ the techniques of global-local shrinkage to problems of fundamental interest in statistics, such as regression, nonlinear function estimation and covariance estimation. More specifically, the PIs aim to show in regression problems that using global-local shrinkage instead of purely global shrinkage methods such as ridge regression or principal components regression can result in improved prediction. They aim to show global-local shrinkage priors are good candidates for non-informative analysis of low-dimensional functions of high-dimensional parameters. The PIs also propose to use global-local shrinkage in covariance estimation and joint mean-covariance estimation problems and apply the developed methodology in suitable applications arising from genomics or finance.