Prior Calibration and Algorithmic Guarantees under Parameter Restrictions

许多真实的系统的统计学习可以通过利用领域知识并将其转化为有意义的参数约束来显著增强。随着高吞吐量数据集的出现，这种限制往往存在于高维参数空间，从而使推理复杂化。本研究的目的是开发新的统计方法和计算算法，这些问题的动机，从一些真实的应用。在非参数贝叶斯框架内工作，研究项目的第一部分强调了在这些约束问题中校准先验分布的重要性，并从理论上量化了约束对参数学习的影响。第二部分主要研究马尔可夫链蒙特卡罗算法和变分算法，并分析其收敛性。PI还将提出本科课程，重点是贝叶斯方法的建模和应用组件。在教授课程时，PI将使用日常生活以及不同学科的科学例子来启发学生的学习。基于活动的学习（ABL）课程旨在丰富学生的学术经验和学习成果，通过将理论与实践，概念与方法联系起来，使用&通过参与更大的世界获得的数据见解。该研究项目的动机是由一些真实的科学应用所带来的统计和计算挑战，其中对关键参数提出了各种复杂的限制，这需要新的统计方法和相关的计算算法。在贝叶斯范式，使纳入各种约束条件的原则框架，并提供现成的不确定性估计往往在科学应用中寻求后，一个主要的重点将放在这些约束空间下的先验分布的校准。将提供一些例子，在实践中经常使用的看似无害的先验选择在某些特定情况下可能导致有偏见的推论。沿着这些约束空间上的替代缺省先验的发展，将提供对这种现象的严格的理论理解。方法和理论的发展将伴随着有效的计算算法，使用新的近似技术的背景下，马尔可夫链蒙特卡罗和变分算法，满足特定应用程序和超越所需的可扩展性。算法的发展将通过新颖的收敛分析，在优化和采样文献之间架起桥梁。该奖项反映了NSF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。

Statistical learning of many real systems can be significantly enhanced by harnessing and translating domain knowledge into meaningful parameter restrictions. With the advent of high throughput datasets, such restrictions are often present on high-dimensional parameter spaces thereby complicating inference. This research aims to develop novel statistical methods and computational algorithms for such problems drawing motivation from a number of real applications. Working within a nonparametric Bayes framework, the first part of the research project lays emphasis on the importance of calibrating prior distributions in these constrained problems and theoretically quantifying the impact of the constraints on parameter learning. The second part aims to develop efficient Markov Chain Monte Carlo and variational algorithms and analyze their convergence behaviors for the said problems. The PIs will also propose undergraduate courses that will focus on the modeling and applied components of Bayesian methods. When teaching the courses, the PIs will use daily life as well as scientific examples across different disciplines to inspire students' learning. The Activity-Based Learning (ABL) courses aim to enrich students' academic experience and learning outcomes by connecting theory with practice and concepts with methods, using data & insights obtained through engagement with the larger world.The research project is motivated by statistical and computing challenges posed by a number of real scientific applications where various complex restrictions are posed on key parameters, necessitating novel statistical methods and associated computational algorithms. Operating in a Bayesian paradigm which enables incorporation of various constraints in a principled framework and provides readily available uncertainty estimates often sought after in scientific applications, a major emphasis will be laid on calibration of prior distributions under these constrained spaces. Examples will be provided where seemingly innocuous prior choices routinely used in practice can lead to biased inferences in certain specific situations. A rigorous theoretical understanding of such phenomenon will be provided along with development of alternative default priors on these constrained spaces. The methodological and theoretical developments will be accompanied by efficient computational algorithms using novel approximation techniques in the context of Markov chain Monte Carlo and variational algorithms that meet the scalability demanded by the specific applications and beyond. The algorithm development will be paralleled by novel convergence analysis, bridging ideas between the optimization and sampling literature.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.