Exchangeable Laws in Borel Data Structures

TL;DR

This paper develops a categorical framework for exchangeability in Borel data structures, generalizing de Finetti's theorem and providing functional representation theorems that unify and extend existing exchangeability results.

Contribution

It introduces Borel data structures and an indexing system, generalizes de Finetti's theorem, and offers explicit functional representation theorems using category theory.

Findings

Generalized de Finetti's theorem for Borel data structures

Unified framework for exchangeability using category theory

Explicit functional representations matching known exchangeable array results

Abstract

Motivated by statistical practice, category theory terminology is used to introduce Borel data structures and study exchangeability in an abstract framework. A generalization of de Finetti's theorem is shown and natural transformations are used to present functional representation theorems (FRTs). Proofs of the latter are based on a classical result by D.N.Hoover providing a functional representation for exchangeable arrays indexed by finite tuples of integers, together with an universality result for Borel data structures. A special class of Borel data structures are array-type data structures, which are introduced using the novel concept of an indexing system. Studying natural transformations mapping into arrays gives explicit versions of FRTs, which in examples coincide with well-known Aldous-Hoover-Kallenberg-type FRTs for (jointly) exchangeable arrays. The abstract "index arithmetic" presented unifies and generalizes technical arguments commonly encountered in the literature on exchangeability theory. Finally, the category theory approach is used to outline how an abstract notion of seperate exchangeability can be derived, again motivated from statistical practice.