Local exchangeability

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 probabilistic models and efficient inference algorithms. In practice, however, this assumption is too strong an idealization; the distribution typically fails to be exactly invariant to permutations and de Finetti's representation theory does not apply. Thus there is the need for a distributional assumption that is both weak enough to hold in practice, and strong enough to guarantee a useful underlying representation. We introduce a relaxed notion of local exchangeability---where swapping data associated with nearby covariates causes a bounded change in the distribution. Next, we prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process, showing that de Finetti's theorem is robust to perturbation and providing further justification for the Bayesian modelling approach. We also provide an investigation of approximate sufficiency and sample continuity properties of locally exchangeable processes on the real line. The paper concludes with examples of popular statistical models that exhibit local exchangeability.