Advances in Bayesian Nonparametric Methods for Jointly Modeling Multiple Data Sets

贝叶斯非参数是一种将经典非参数统计方法的灵活性与贝叶斯范式下的不确定性原则评估相结合的统计建模框架。然而，传统的贝叶斯非参数方法主要集中在基于单个数据集的模型上，而许多现代统计场景涉及在相关或比较条件下收集的性质相似的多个数据集。该项目开发了一套新的建模和计算策略，为多个数据集的有效联合建模量身定制，其方式(i)捕获现代复杂数据中的跨样本变化，以及（ii）计算效率高，允许应用于大量数据。所开发的方法将对一系列领域产生影响，包括生物学、经济学、教育学、天体物理学、政治学和气候科学，这些领域的任务是正确地描述数据集之间的变化。本项目为研究生提供了良好的科研训练机会。新的模型、方法和算法将在两类广泛使用的贝叶斯非参数模型的背景下开发：(i)具有离散随机测量（DRM）混合分布的混合模型（例如，Dirichlet过程混合）和（ii）树结构随机测量（TSRM）模型（例如，Polya树型模型）。这两个模型类本质上是不同的，在对多个数据集建模时，每个都有自己的优点和局限性，因此推进这两个模型类的策略是不同的。DRM混合模型在多样本建模方面的一个关键限制是其在表征跨样本变化方面缺乏灵活性，因此需要开发一种新的潜在变量建模策略，以大大提高其在这方面的能力。还将研究由此产生的分散混合模型的理论和经验性质，以及在更广泛的层次模型范围内推广该策略，其中在观察量和潜在量中纳入灵活的跨样本变化是很重要的。对于能够表征复杂的跨样本变化的TSRM模型，重点是解决它们在计算和统计方面缺乏可扩展性的问题，增加维度以及它们对底层树结构的敏感性，这是构建此类模型的关键组成部分。这两个模型类的开发将形成一个强大而通用的工具箱，可以应用于涉及多个相关数据集分析的各种科学和工程问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Bayesian nonparametrics is a statistical modeling framework that combines flexibility of classical nonparametric statistical methods with principled assessment of uncertainty under the Bayesian paradigm. Traditional Bayesian nonparametric methods, however, have largely focused on models based on a single data set, while many modern statistical scenarios involve multiple data sets of similar nature collected under related or comparative conditions. This project develops a suite of new modeling and computational strategies that are tailored for effective joint modeling of multiple data sets in ways that (i) capture cross-sample variation in modern complex data, and (ii) are computationally efficient to allow application to massive data. The developed methodology will have impact in a range of fields including biology, economics, education, astrophysics, political science, and climate science, where the task is to properly characterize variation across data sets. The project provides excellent research training opportunities for graduate students.Novel models, methods, and algorithms will be developed in the context of two classes of widely used Bayesian nonparametric models: (i) mixture models with discrete random measure (DRM) mixing distributions (e.g., Dirichlet process mixtures) and (ii) tree-structured random measure (TSRM) models (e.g., Polya tree type models). These two model classes are different in nature with each having its own advantages and limitations in modeling multiple data sets, and as such the strategies in advancing these two model classes are distinct. A key limitation of DRM mixtures in modeling multiple samples is their lack of flexibility in characterizing the cross-sample variation, and thus a new latent variable modeling strategy will be developed for substantially enhancing their capacity in this regard. Also to be investigated are the theoretical and empirical properties of the resulting dispersion mixture models, as well as generalizations of this strategy in broader ranges of hierarchical models where incorporating flexible cross-sample variation in observed and latent quantities is important. For TSRM models, capable of characterizing complex cross-sample variation, the focus is on addressing their lack of scalability, both computational and statistical, with respect to increasing dimensionality as well as their sensitivity to the underlying tree structures, a critical component for building such models. The development for these two model classes will form a powerful and general toolbox that can be applied in a variety of scientific and engineering problems involving the analysis of multiple related data sets.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.