CAREER: Geometric approaches to hierarchical and nonparametric model-based inference

层次和非参数模型是现代统计学中最基本和最强大的工具。尽管在过去的几十年里取得了宝贵的进展，但仍有几个公认的新问题。首先，尽管这些模型越来越多地应用于大型数据集和复杂领域，但模型中存在的潜在变量层次的推断行为的统计理论尚不可用。其次，基于采样的局部推理方法虽然推导简单，但往往收敛速度太慢，从而失去有效性。第三，大多数现有方法都无法处理高度分布式的数据源，而高度分布式的数据源正日益成为大数据涌入的原因。解决这些挑战需要从根本上在理论、建模和算法方面有新的想法，这些想法必须考虑到推理问题的全局几何形状与推理和算法实现的去中心化需求之间的对比和相互作用。该项目旨在为推进基于分层模型的推理做出根本性贡献。它们包括变量潜在层次的统计理论和迁移学习效果的分析。它们还包括基于潜在结构的全局几何和统计和计算效率之间权衡的几何分析的变分推理算法。算法和理论都是通过使用沃瑟斯坦几何来统一的，沃瑟斯坦几何源于最优运输的数学理论。此外，将开发可扩展的分层模型，可以利用高度分布式的数据源和分散的推理体系结构。这项研究将提高我们对大规模、高维、复杂数据的管理、分析和决策能力，特别是在网络和环境的研究和应用中。高度分布式数据源的分散检测算法有可能推动支持数据驱动和高性能分布式计算架构的最新技术。因此，这项研究有可能扩展目前在医疗保健和安全领域部署的实时检测和跟踪设备的功能。基于最优运输的理论将加深我们对层次贝叶斯推理的理解，这是现代统计学的一个基本概念。算法和几何分析将在统计复杂性和计算复杂性之间提供有用的权衡，这是统计学和计算机科学之间的一个重要问题。这项研究还将为扩大密歇根大学目前的统计学课程提供支持。PI将通过开发与项目研究主题密切互动的综合课程，将统计和计算工具的教学与现代应用相结合。这为培养学生在统计学、概率论、机器学习、分布式计算和网络系统等领域的广泛知识和跨学科技能提供了极好的机会。

Hierarchical and nonparametric models present some of the most fundamental and powerful tools in modern statistics. Despite valuable advances made in the past decades, there are several widely recognized and emerging problems. First, even as these models are increasingly applied to large data sets and complex domains, a statistical theory for inferential behaviors of the hierarchy of latent variables present in the models is not yet available. Second, local inference methods based on sampling, although simple to derive, tend to converge too slowly, thereby losing their effectiveness. Third, most existing methods are incapable of handling highly distributed data sources, which are increasingly responsible for the influx of big data. Addressing these challenges requires fundamentally new ideas in theory, modeling and algorithms that must account for the contrast and interplay between the global geometry of an inference problem and the need for decentralization of inference and algorithmic implementation. This project aims to make fundamental contributions toward advancing hierarchical model-based inference. They include a statistical theory for the latent hierarchy of variables and for analyzing the effects of transfer learning. They also include variational inference algorithms based on the global geometry of latent structures and geometric analyses of the tradeoffs between statistical and computational efficiencies. Both the algorithms and theory are unified by the use of Wasserstein geometry, which arises from the mathematical theory of optimal transportation. Moreover, scalable hierarchical models will be developed that can exploit highly distributed data sources and decentralized inference architectures.This research will improve our ability to manage, analyze and make decisions with large-scale, high dimensional and complex data, especially in the research and applications of networks and the environment. The decentralized detection algorithms for highly distributed data sources have the potential of advancing the state of the art technologies that support data-driven and high-performance distributed computing architectures. As such, this research has the potential of extending the capabilities of the real-time detection and tracking devices currently deployed in the health-care and security domains. The optimal transport based theory will deepen our understanding of hierarchical Bayesian inference, a fundamental concept of modern statistics. The algorithms and geometric analyses will provide useful tradeoffs between statistical and computational complexity, an important issue lying in the interface of Statistics and Computer Science. This research will also provide support for broadening the current statistics curriculum at the University of Michigan. The PI will integrate the teaching of statistical and computational tools with modern applications, by developing synthesis courses which interact closely with research topics of the project. This provides an excellent opportunity to train students with a broad base of knowledge and cross-disciplinary skills in the fields of statistics, probability, machine learning, distributed computation and networked systems.