Randomised Polya tree models for nonparametric Bayesian inference

By comparison with modern parametric Bayesian statistics, practicable and robust methods for exploration and data analysis in nonparametric settings are underdeveloped. The rapid development of non-Bayesian methods and ranges of ad-hoc non-parametric tools for data mining reflect the need for a non-parametric Bayesian approach to exploring and managing data sets in even moderate dimensional problems. We address this issue by presenting multivariate Polya tree based methods for modelling multidimensional probability distributions. To address the issue of "partition dependence" -- an outstanding limitation of Polya trees and other partition-based models -- we develop Randomised Polya Trees. This new framework inherits the attractive hierarchical, multiscale structure of Polya trees but "rubberises" partition points and as a result smooths away discontinuities in predictive distributions. Some of the theoretical aspects of the new framework are developed, followed by discussion of methodological and computational issues arising in implementation. Examples of data analyses and prediction problems are provided to highlight issues of Bayesian learning in this context, and to indicate future research directions.