Bayesian Nonparametric Approaches To Multiple Testing, Density Estimation, And Supervised Learning

This dissertation presents methods for several applications of Polya tree models. These novel nonparametric approaches to the problems of multiple testing, density estimation and supervised learning provide an alternative to other parametric and nonparametric models. In Chapter 2, the proposed approximate finite Polya tree multiple testing procedure is very successful in correctly classifying the observations with non-zero mean in a computationally efficient manner; this holds even when the non-zero means are simulated from a mean-zero distribution. Further, the model is capable of this for “interestingly different” observations in the cases where that is of interest. Chapter 3 proposes discrete, and smoothed approximate mixtures of Polya trees for application in mixed models and density estimation. Finally, Chapter 4 proposes a supervised learning procedure based on marginal, multivariate finite Polya trees. This approach is successful in correctly classifying observations in a variety of scenarios where the ten-fold cross validation was kept low. The proposed methodologies and applications show the versatility and flexibility of nonparametric Polya tree based methods and Chapter 5 outlines some obvious but rich extensions for future research