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.
T. Campbell is supported by a National Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant and Discovery Launch Supplement. T. Broderick is supported in part by an NSF CAREER Award, an ARO YIP Award, ONR, and a Sloan Research Fellowship.
The authors thank Jonathan Huggins for illuminating discussions.
Trevor Campbell. Saifuddin Syed. Chiao-Yu Yang. Michael I. Jordan. Tamara Broderick. "Local exchangeability." Bernoulli 29 (3) 2084 - 2100, August 2023. https://doi.org/10.3150/22-BEJ1533