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

大多数科学结论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，可能会导致不正确或误导性的结果。在许多现代应用中，统计建模因总体固有的结构不均而变得更加复杂。具体的例子包括信息在社会网络中的传播、来自不同物种的线粒体DNA序列之间的遗传差异、悬浮在非均匀介质中的颗粒的碰撞，以及在犯罪或恐怖网络中发现可疑活动。考虑到这些应用，该项目启动了对来自不同种类或特殊结构人口的数据的概率模型和推断原则的系统研究。实现这些目标需要来自几个数学和科学领域的技术，并应导致在应用科学、统计学、概率和数学的交叉领域取得进展。项目成果将有助于更好地理解基本统计假设如何影响科学结论的有效性。所发展的数学理论和统计方法应该产生广泛的社会影响，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得到了广泛的应用。PI制定了广泛的计划，通过两个级别的统计学基础课程、博士论文的指导以及为这些研究人员和其他职业生涯早期研究人员举办的关于这些主题的会议和讲习班，对研究生和本科生进行使用方法的培训。这些计划是创新的，PI将在这些计划上投入大量时间和资源。特别强调吸引和聘用来自贫困家庭的学生，他们是第一代大学生。具体的技术目标是理解随机结构的严格数学理论，这些结构表现出相对可交换性和其他不变性原理。该项目的一个标志是对边可交换网络模型的理论和应用进行了深入的研究和严格的发展，该模型最初是由PI作为一种新的网络分析不变性原理引入的。期望的结果包括进化组合结构的相对不变性原理、结构性质和刻画定理。该项目还将利用组合学、代数、逻辑和概率论之间的深层联系来完善De Finetti、Kingman、Aldous、Hoover和Kallenberg在组合状态空间上关于图极限和Feller过程的Levy-Ito型表示的先前工作。理论发展应指导方法学在具体应用方面的进步，包括气候科学、网络科学和系统发育学。

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.