Tree-based Methods for Learning Probability Distributions

Abstract

Learning probability distributions is a fundamental inferential task in statistics but challenging if a data distribution of our interest is complicated and high-dimensional. Addressing this challenging problem is the main topic of this thesis, and mainly discussed herein are two types of new tree-based methods: a single-tree method and an ensemble method. The new single tree method, the main topic of Chapter 2, is introduced by constructing a generalized Polya tree process, that is, a new Bayesian nonparametric model, equipped with a new flexible tree prior. With this new prior we can find trees that represent the distributional structures well, and the tree space is efficiently explored with a new sequential Monte Carlo algorithm. The new ensemble method discussed in Chapter 3 is proposed under a new addition rule defined for probability distributions. The new rule based on cumulative distribution functions and their generalizations enables us to smoothly introduce a new efficient boosting algorithm, inheriting the important notions such as "residuals" and "zeros"..The thesis is closed by Chapter 4 which provides concluding remarks.