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

Autor: Carlier, Guillaume, Friesecke, Gero, Vögler, Daniela
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Zdroj: Probability Theory & Related Fields; Feb2023, Vol. 185 Issue 1/2, p311-351, 41p
Abstrakt: 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 μ k of the first k components has a representation of the form μ k = ∫ P 1 N (X) F N , k (λ) d α (λ) for some probability measure α on the set of 1 N -quantized probability measures on X and certain universal polynomials F N , k . The latter consist of a leading term N k - 1 / ∏ j = 1 k - 1 (N - j) λ ⊗ k and a finite, exponentially decaying series of correlated corrections of order N - j ( j = 1 , ... , k ). The F N , k (λ) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals λ . 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. [ABSTRACT FROM AUTHOR]
Databáze: Complementary Index
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