Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections

Abstract

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If \((Z_1, \ldots ,Z_N)\) is a finitely exchangeable sequence of N random variables taking values in some Polish space X, we show that the law \(\mu _k\) of the first k components has a representation of the form $$\begin{aligned} \mu _k=\int _{\mathcal{P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda ) \, \text{ d } \alpha (\lambda ) \end{aligned}$$for some probability measure \(\alpha \) on the set of \(\frac{1}{N}\)-quantized probability measures on X and certain universal polynomials \(F_{N,k}\). The latter consist of a leading term \(N^{k-1}\! /\prod _{j=1}^{k-1}(N\! -\! j) \lambda ^{\otimes k}\) and a finite, exponentially decaying series of correlated corrections of order \(N^{-j}\) (\(j=1, \ldots ,k\)). The \(F_{N,k}(\lambda )\) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals \(\lambda \). Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis and Freedman (Ann Probab 8(4):745–764, 1980) between finite and infinite exchangeable laws.