Bayesian Recursive Partitioning and Inference on the Structure of High-Dimensional Distributions

本研究涉及现代数据分析中最重要和最普遍的一类问题-概率分布结构的推断。要解决的具体推理问题分为两大类。第一个涉及到一个单一的概率分布的结构，包括估计联合和条件密度，在线性回归的变量选择，以及测试的独立性和条件独立性变量的推断。第二个涉及对多个分布之间关系的推断。这包括测试两个（或多个）数据样本是否具有相同的基本分布，并学习它们的差异结构，特别关注在大型高维空间中找到局部结构--位于小子集中的差异。为了解决这些问题，研究者提出了一个新的框架，通过递归划分构建贝叶斯先验的多元分布。使用该框架的推理是灵活的和自适应的。此外，这些先验的生成性质有利于跨多个分布的依赖结构的建模，这导致了用于比较分布的强大方法。为了解决高维问题中的计算挑战，研究者提出了一套计算策略，并提出了几种算法，可以大大提高贝叶斯后验推理在高维问题中的效率。这些策略利用所提出的框架的递归性质，有效地探索相应的后验分布的全局景观。概率分布结构的推断是许多科学研究的核心，迫切需要新的统计理论和方法来适应现代科学调查中常见的不断增加的数据集维数。激发该项目的两个具体应用是分析系统生物学中的高维流式细胞术数据，以揭示蛋白质之间的功能关系，以及将人类基因映射到各种定性和定量特征，特别是癌症和糖尿病等常见疾病。在这个项目中开发的概念，理论，方法和算法将直接适用于这些问题，以及从环境科学到经济学的各种其他领域所产生的数据集的分析。

This research concerns one of the most important and pervasive classes of problems in modern data analysis---inference on the structure of probability distributions. Specific inference problems to be addressed fall into two broad categories. The first involves inference on the structure of a single probability distribution, including estimation of joint and conditional densities, variable selection in linear regression, and the testing of independence and conditional independence among variables. The second involves inference on the relationship across multiple distributions. This includes testing whether two (or more) data samples have the same underlying distribution, and learning the structure of their difference, with particular interest given to finding local structures---differences that lie in small subsets---in large high-dimensional spaces. To address these problems, the investigator puts forward a novel framework for constructing Bayesian priors on multivariate distributions through recursive partitioning. Inference using this framework is flexible and adaptive. Moreover, the generative nature of these priors facilitates the modeling of dependence structure across multiple distributions and this leads to powerful methods for comparing distributions. To address the computational challenges in high-dimensional problems, the investigator lays out a set of computational strategies and proposes to develop several algorithms that can drastically improve the efficiency of Bayesian posterior inference in high-dimensional problems. These strategies utilize the recursive nature of the proposed framework to efficiently explore the global landscape of the corresponding posterior distributions.Inference on the structure of probability distributions lies at the heart of many scientific inquiries, and new statistical theory and methods are urgently needed to accommodate the ever increasing dimensionality of data sets that is commonplace in modern scientific investigations. Two specific applications that motivate this project are the analysis of high-dimensional flow cytometry data in systems biology for unraveling the functional relationships among proteins as well as the mapping of human genes to various qualitative and quantitative traits, in particular those of common diseases such as cancer and diabetes. The concepts, theory, methodology, and algorithms developed in this project will be directly applicable to these problems, as well as to the analysis of data sets arising from a wide variety of other fields ranging from environmental science to economics.