High Dimensional Bayesian Model Discovery, Inference and Prediction

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

This research is concerned generally with new methods for regression analysis in high dimensions, where the number of parameters is very large compared to the number of observations. The first part of this research focuses on the development and application of methods that identify and draw inference about low dimensional structure in this setting. For this purpose, the investigator further develops a new nonparametric effects model called BART (Bayesian Additive Tree Models) that uses a dimensionally adaptive dynamic random basis of trees. In particular, BART opens up a variety of new ways to perform variable selection followed by model selection. As opposed to traditional variable selection methods that rely on a preselected parametric model class, these new methods will select variables before the selection of a model, thus eliminating a major limitation of previous methods. The second part of this research focuses on the development of new Bayesian procedures for predictive density estimation in multiple regression. The minimaxity of such density estimates under Kullback-Leibler loss are established for various classes of scaled superharmonic priors. Based on these estimates, the investigator further develops minimax multiple shrinkage predictive density estimates that exploit variable selection uncertainty to achieve risk reduction. The discovery of structure in complex settings and the prediction of future uncertain events are fundamental statistical challenges in most areas of academic research including business, the hard and social sciences, medicine, humanities, computer science and engineering. By modeling and predicting economic conditions, the economist can better direct the economy; by modeling and predicting climate changes, the scientist can better manage the environment; by modeling and predicting health care needs, the policy analyst can better allocate resources to meet demand. This research develops brand new procedures that establishes high dimensional model discovery and predictive estimation as highly visible and valuable areas of theoretical and methodological research, one that will attract the brightest students and the wisest seasoned scholars. This will entail broad dissemination of the work in pinnacle journals, through public lectures around the world, and through new collaborations with colleagues and graduate students. To further facilitate the use of these new methods, the investigator will continue to create and make publicly available open-source software implementations along with full documentation and examples.