Scalable Bayesian regression: Analytical and numerical tools for efficient Bayesian analysis in the large data regime

分层回归已成为统计学和数据科学中普遍使用的工具。社会科学和自然科学中的应用研究人员，在流行病学、政治学、基因组学等领域，依靠分层回归作为其数据分析工具箱的基本元素。然而，分层回归在实际应用和广泛应用方面仍然存在重大障碍。主要的限制是计算-从具有大量数据的复杂模型中学习可能需要大量的计算时间。通过这一研究项目，研究人员的目标是为应用统计学家和数据科学家社区提供一系列工具，以开启更高效、用户友好的统计建模。该项目为学生提供研究培训机会。研究人员将重点开发用于分层建模的统计推断的定制计算和分析工具。常用的推理工具，如MCMC和变分方法，很少利用模型中友好的分析结构，它们通常依赖于对目标密度及其梯度的许多评估。在这个项目中，研究人员将利用后验密度的分析和数值特性，并通过建立定制的方法来利用模型往往广泛友好的结构。这将涉及到使用现代数值线性代数、逼近理论和快速算法工具来数值求解偏微分方程组。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Hierarchical regression has become a ubiquitous tool in statistics and data science. Applied researchers across the social and natural sciences, in fields such as epidemiology, political science, genomics, and many more, rely on hierarchical regression as an essential element of their data analysis toolbox. However, there are still major obstacles in practical and widespread use of hierarchical regression. The primary limitation is computational—learning from complex models with large amounts of data can require extensive compute time. With this research project, the investigators aim to supply the communities of applied statisticians and data scientists with a range of tools that open up more efficient user-friendly statistical modeling. This project provides research training opportunities for students.The investigators will focus on the development of customized computational and analytical tools for statistical inference for hierarchical modeling. Popular tools for inference such as MCMC and variational methods rarely take advantage of friendly analytical structure in the model and they typically rely on many evaluations of the target density and its gradient. In this project, the investigators will exploit the analytical and numerical properties of the posterior density and capitalize on the oftentimes extensive friendly structure of models by building customized methods. This will involve the use of modern numerical linear algebra, approximation theory, and the tools of fast algorithms for the numerical solution of partial differential equations.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.