Exchangeability Theory of ID-Based Data Structures with Applications in Statistics

• Applicant: Dr. Julian Gerstenberg
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 502386356

Exchangeability is a probabilistic symmetry of fundamental relevance to statistics as it is a reasonable assumption on the statistical distribution of data gathered by picking individuals totally random from a large population. A first and famous result of exchangeability theory (de Finetti/Hewitt-Savage) says that exchangeable sequences equal (in distribution) mixtures of iid sequences, where iid = independent and identically distributed. Besides sequential data, many other data structures have an interesting notion of exchangeability. The long and growing list of data structures whose exchangeability theories have been considered in the literature includes partitions, graphs, orders, arrays, hierarchical structures and more. The proposed research project aims to develop a general theory of exchangeability in data structures. The approach is to introduce an abstract notion of data structure using the language of category theory. This approach allows to introduce an abstract notion of exchangeability in such a way that all known examples of exchangeability in data structures are instances of this new approach. The study of the literature of many concrete examples of data structures shows that exchangeability shares close connections to other topics in mathematics. Within the newly developed framework this connections can be explored in general. One main goal of the proposed project is a unified and generalized formulation of the famous Aldous-Hoover-Kallenberg representation theorems. Applications of the proposed reserach project are in statistics and in particular in empirical risk minimization principles, also extensively used in the field of Machine learning. Other applications are thinkable in computer science, as the developed abstract framework for data structures directly guides the way towards the design of software packages providing several kinds of data structures.

• DFG Programme: Research Grants