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

• 成果类型: Article
• 署名作者: Carlier, Guillaume; Friesecke, Gero; Voegler, Daniela
• 署名单位: Universite PSL; Universite Paris-Dauphine; Technical University of Munich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01115-2
• 发表日期: 2023
• 页码: 311-351
• 关键词: optimal transport; sampling theory; THEOREM

摘要

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), ..., 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 mu(k )= integral F-P1/N(X) (N,k)(lambda) d alpha(lambda) for some probability measure alpha on the set of 1/N-quantized probability measures on X and certain universal polynomials F-N,F-k. The latter consist of a leading term Nk-1/Pi(k-1)(j=1)(N-j)lambda(circle times k) and a finite, exponentially decaying series of correlated corrections of order N-j (j = 1, ..., k). The F-N,F-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.