CAREER: Flexible and Efficient Exploration of the Bayesian Framework for High Dimensional Modeling

现代大数据时代给统计学家带来了独特的机遇和挑战。虽然大数据革命为从丰富的数据中获得有价值的深刻见解和加强数据驱动的决策带来了巨大的机会，但它也在三个关键方面对统计学家和数据科学家的创新和知识发现提出了挑战性的要求：(i)开发能够适当描述数据复杂性的灵活模型（ii）高效和有效的统计估计和推理程序，以及（iii）开发可扩展到大型数据集的计算算法。本项目的目的是通过充分探索贝叶斯框架，在这三个方面都取得进展，贝叶斯框架将模型参数视为随机的，并提供了一种有效的机制来量化模型参数的不确定性。特别是，所开发的技术将有助于分析包含大量协变量的数据集，学习大量结果变量之间的依赖结构，以及通过在不同分位数水平上建模协变量对结果变量的影响来全面描述协变量对结果变量的影响。这项研究将对包括生物学、经济学、环境科学、市场营销和医学在内的各个学科的统计实践产生影响。培训部分将通过为研究生提供基于拟议研究的专题课程，以及通过开发将研究概念纳入无障碍水平的本科生研究项目，将研究纳入教学。PI将指导高中研究项目，并组织一个K-12外展讲习班，向高中学生和教师提供现代统计学及其应用的机会。统计严谨和计算高效的贝叶斯方法和推理程序将被开发，这将适用于各种复杂的高维模型，包括广义线性模型，分位数回归模型和图形模型。本文将提出贝叶斯正则化先验的一般类别，并对各种常用的似然函数严格研究其正则化性质。与大多数现有的专注于高维估计的贝叶斯方法相反，将开发一种新的贝叶斯框架，用于执行具有有效频率性的高维贝叶斯推理。不涉及大型矩阵操作的可扩展计算技术，用于从后验中获得点估计量以及对完整后验分布进行采样，并将研究其统计特性。计算发展的一个吸引人的特点是，它们将适用于实践中常用的各种统计模型。所开展的研究将与现代统计的几个高度活跃的领域密切相关，包括高维建模、贝叶斯计算、非凸正则化、后选择推理、图形模型和分位数回归，并将有助于这些领域的进步和相互作用。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。

The modern era of Big Data brings unique opportunities as well as challenges to the statistician. While the Big Data revolution brings a great opportunity to obtain valuable and profound insights from the richness of data and to enhance data-driven decision making, it also brings challenging demands for innovation and knowledge discovery in three crucial aspects from statisticians and data scientists: (i) development of flexible models that can appropriately describe the complexities of the data (ii) efficient and valid statistical estimation and inferential procedures, and (iii) development of computational algorithms that scale-up to large datasets. The purpose of this project is to make advances in all the three aspects by fully exploring the Bayesian framework, which treats the parameters of a model to be random and provides an efficient mechanism to quantify the uncertainty of the model parameters. In particular, the techniques developed will be useful for analyzing datasets containing a large number of covariates, for learning the dependence structures between a large number of outcome variables, and for obtaining a comprehensive description of the impact of covariates on outcome variables by modeling their relationships at different quantile levels. The research developed will have impact on statistical practice in various disciplines including biology, economics, environmental sciences, marketing, and medical sciences. The training component will integrate research into teaching by offering special topics courses to graduate students based on the proposed research and by developing undergraduate research projects that incorporate research concepts at an accessible level. The PI will mentor high school research projects and organize a K-12 outreach workshop to provide exposure to modern statistics and its applications to high school students and teachers. Statistically rigorous and computationally efficient Bayesian methodologies and inferential procedures will be developed which will be applicable for a variety of complex high dimensional models including generalized linear models, quantile regression models, and graphical models. General classes of Bayesian regularization priors will be proposed, and their regularization properties will be rigorously studied for a variety of commonly used likelihood functions. In contrast to most of the existing Bayesian approaches that focus on high dimensional estimation, a novel Bayesian framework for performing high dimensional Bayesian inference having valid frequentist properties will be developed. Scalable computational techniques that do not involve large matrix operations for obtaining point estimators from the posteriors as well as for sampling the full posterior distributions will be devised and their statistical properties will be studied. An attractive feature of the computational developments will be that they will be applicable to a diverse range of statistical models commonly used in practice. The research developed will be closely related to several highly active areas of modern statistics including high dimensional modeling, Bayesian computation, nonconvex regularization, post-selection inference, graphical models, and quantile regression, and will contribute to the advancement of and interaction between these areas.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.