# Local Exchangeability

## TL;DR

This paper introduces the concept of local exchangeability, a relaxation of exchangeability allowing bounded distributional changes under local data swaps, with implications for Bayesian nonparametrics and permutation tests.

## Contribution

It formalizes local exchangeability, proves its connection to measure-valued processes, and demonstrates practical applications in Bayesian inference and hypothesis testing.

## Key findings

- Local empirical measures approximate underlying processes.
- Local exchangeability characterizes certain stochastic processes.
- Applications include Bayesian nonparametrics and covariate-dependent tests.

## Abstract

Exchangeability -- in which the distribution of an infinite sequence is invariant to reorderings of its elements -- implies the existence of a simple conditional independence structure that may be leveraged in the design of statistical models and inference procedures. In this work, we study a relaxation of exchangeability in which this invariance need not hold precisely. We introduce the notion of local exchangeability -- where swapping data associated with nearby covariates causes a bounded change in the distribution. We prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process. Using this main probabilistic result, we show that the local empirical measure of a finite collection of observations provides an approximation of the underlying measure-valued process and Bayesian posterior predictive distributions. The paper concludes with applications of the main theoretical results to a model from Bayesian nonparametrics and covariate-dependent permutation tests.