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的统计分析。反过来，统计分析依赖于假设，如果违反假设，可能导致不正确或误导性的结果。在许多现代应用中，人口中固有的结构不均匀性使统计建模进一步复杂化。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序列的遗传变异，悬浮在非均匀介质中的粒子碰撞，以及犯罪或恐怖网络中可疑活动的识别。考虑到这些应用程序，该项目启动了对从异质或特殊结构人群中获取的数据的概率模型和推理原则的系统研究。实现这些目标需要来自多个数学和科学领域的技术，并应在应用科学、统计学、概率论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都得到了广泛的应用。PI有广泛的计划，通过两个层次的统计学基础课程，博士论文的监督，以及为这些和其他早期职业研究人员举办的关于这些主题的会议和讲习班，来训练研究生和本科生使用的方法。这些计划具有创新性，PI将为此投入大量时间和资源。特别是，将把重点放在吸引和聘用第一代大学生出身的贫困阶层学生上。具体的技术目标是一个严谨的数学理论，用于理解表现出相对互换性和其他不变性原则的随机结构。该项目的一个标志是对边缘交换网络模型的理论和应用的深入研究和严格发展，这是PI首次作为网络分析的一种新的不变性原则引入的。期望的结果包括相对不变性原理、结构性质和演化组合结构的表征定理。该项目还将探索组合学、代数、逻辑和概率论之间的深层联系，以完善de Finetti、Kingman、Aldous、Hoover和Kallenberg先前关于组合状态空间上Feller过程的图极限和levy - ito型表示的工作。理论发展应该指导具体应用的方法进步，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.