<p>Two-sample comparison is a fundamental problem in statistics. Given two samples of data, the interest lies in understanding whether the two samples were generated by the same distribution or not. Traditional two-sample comparison methods are not suitable for modern data where the underlying distributions are multivariate and highly multi-modal, and the differences across the distributions are often locally concentrated. The focus of this thesis is to develop novel statistical methodology for two-sample comparison which is effective in such scenarios. Tools from the nonparametric Bayesian literature are used to flexibly describe the distributions. Additionally, the two-sample comparison problem is decomposed into a collection of local tests on individual parameters describing the distributions. This strategy not only yields high statistical power, but also allows one to identify the nature of the distributional difference. In many real-world applications, detecting the nature of the difference is as important as the existence of the difference itself. Generalizations to multi-sample comparison and more complex statistical problems, such as multi-way analysis of variance, are also discussed.</p> / Dissertation

This thesis explores stochastic modeling and Bayesian inference strategies in the context of the following three problems: 1) Modeling the complex interactions between and within molecules; 2) Extracting information from stepwise signals that are commonly found in biophysical experiments; 3) Improving the computational efficiency of a non-parametric Bayesian inference algorithm. Chapter 1 studies the data from a recent single-molecule biophysical experiment on enzyme kinetics. Using a stochastic network model, we analyze the autocorrelation of experimental fluorescence intensity and the autocorrelation of enzymatic reaction times. This chapter shows that the stochastic network model is capable of explaining the experimental data in depth and further explains why the enzyme molecules behave fundamentally differently from what the classical model predicts. The modern knowledge on the molecular kinetics is often learned through the information extracted from stepwise signals in experiments utilizing fluorescence spectroscopy. Chapter 2 proposes a new Bayesian method to estimate the change-points in stepwise signals. This approach utilizes marginal likelihood as the tool of inference. This chapter illustrates the impact of the choice of prior on the estimator and provides guidelines for setting the prior. Based on the results of simulation study, this method outperforms several existing change-points estimators under certain settings. Furthermore, DNA array CGH data and single molecule data are analyzed with this approach. Chapter 3 focuses on the optional Polya tree, a newly established non-parametric Bayesian approach (Wong and Li 2010). While the existing study shows that the optional Polya tree is promising in analyzing high dimensional data, its applications are hindered by the high computational costs. A heuristic algorithm is proposed in this chapter, with an attempt to speed up the optional Polya tree inference. This study demonstrates that the new algorithm can reduce the running time significantly with a negligible loss of precision. / Statistics