Exchangeable Processes: de Finetti’s Theorem Revisited

Abstract

A sequence of random variables is exchangeable if the joint distribution of any finite subsequence is invariant to permutations. De Finetti’s representation theorem states that every exchangeable infinite sequence is a convex combination of independent and identically distributed processes. In this paper, we explore the relationship between exchangeability and frequency-dependent posteriors. We show that any stationary process is exchangeable if and only if its posteriors depend only on the empirical frequency of past events.