CAREER: Probabilistic Foundations, Statistical Inference, and Invariance Principles for Evolving Combinatorial Structures

大多数科学结论都依赖于对从某些物理过程中收集的数据进行统计分析。反过来，统计分析依赖于假设，如果违反这些假设，可能导致不正确或误导性的结果。在许多现代应用中，统计建模由于群体中固有的结构不均匀性而进一步复杂化。具体的例子包括信息在社交网络中的传播、不同物种线粒体DNA序列之间的遗传变异、悬浮在不均匀介质中的粒子的碰撞以及识别犯罪或恐怖主义网络中的可疑活动。考虑到这些应用，该项目启动了一个系统的研究概率模型和推理原则的数据从异质或特殊结构的人口。这些目标的实现需要从几个数学和科学领域的技术，并应导致在应用科学，统计，概率和数学的交叉进步。项目成果将使人们更好地了解基本统计假设如何影响科学结论的有效性。开发的数学理论和统计方法应该具有广泛的社会影响，因为组合随机过程在物理，生物和社会科学，国家安全等领域的现代应用中使用。PI有广泛的计划培训研究生和本科生的方法要使用，通过课程的基础上统计在这两个级别，博士的监督。为这些和其他早期职业研究人员举办关于这些主题的论文、会议和研讨会。这些计划是创新的，PI将为其投入大量时间和资源。特别强调吸引和雇用来自贫困背景的第一代大学生。具体的技术目标是理解表现出相对交换性和其他不变性原理的随机结构的严格数学理论。 该项目的一个标志是对边缘可交换网络模型的理论和应用进行深入研究和严格开发，该模型首先由PI作为网络分析的新不变性原理引入。 期望的结果包括相对不变性原则，结构特性，和特征定理的演变组合结构。该项目还将利用组合学，代数，逻辑和概率论之间的深刻联系，以完善de Finetti，金曼，Aldous，Hoover和Kallenberg先前的工作，关于组合状态空间上Feller过程的图极限和Levy-Ito型表示。理论发展应指导具体应用中的方法学进展，包括气候科学、网络科学和遗传学。

Most scientific conclusions rely on statistical analysis of data collected from some physical process. The statistical analysis, in turn, relies on assumptions which, if violated, can lead to incorrect or misleading outcomes. In many modern applications, the statistical modeling is further complicated by inherent structural inhomogeneities in the population. Specific examples include the spread of information in social networks, genetic variation among mitochondrial DNA sequences from different species, collisions of particles suspended in a non-uniform medium, and identification of suspicious activity in a criminal or terrorist network. With these applications in mind, the project initiates a systematic study of probabilistic models and inferential principles for data taken from heterogeneous or specially structured populations. Attainment of these goals requires techniques from several mathematical and scientific areas and should lead to progress at the intersection of applied science, statistics, probability, and mathematics. Project outcomes will lead to a better understanding of how fundamental statistical assumptions affect the validity of scientific conclusions. The mathematical theory and statistical methods developed should have a broad societal impact, as combinatorial stochastic processes are used throughout modern applications in physical, biological, and social sciences, national security, and beyond. The PI has extensive plans for training graduate and undergraduate students in the methods to be used, via courses on the foundations of statistics at both levels, the supervision of Ph.D. theses, and conferences and workshops on these topics for these and other early-career researchers. The plans are innovative and the PI will devote significant time and resources to them. In particular, a strong emphasis will be placed on attracting and hiring students from underprivileged backgrounds who are first-generation college students.The specific technical objective is a rigorous mathematical theory for understanding random structures that exhibit relative exchangeability and other invariance principles. A hallmark of the project is the in-depth study and rigorous development of theory and applications for edge exchangeable network models, which were first introduced by the PI as a novel invariance principle for network analysis. Desired outcomes include relative invariance principles, structural properties, and characterization theorems for evolving combinatorial structures. The project will also exploit deep connections between combinatorics, algebra, logic, and probability theory to refine prior work by de Finetti, Kingman, Aldous, Hoover, and Kallenberg, on graph limits and Levy-Ito-type representations of Feller processes on combinatorial state spaces. Theoretical developments should guide methodological advances in specific applications, including climate science, network science, and phylogenetics.