arXiv 2106.091012021-06-16math.PRmath.OC

有限可交换律的凸几何和具有万有关联修正的de Finetti型表示

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

Guillaume Carlier, Gero Friesecke, Daniela Vögler

摘要：

我们提出了一种新的模拟有限交换序列的de Finetti，Hewitt和Savage定理，并研究其影响的多边际最优运输（MMOT）和贝叶斯统计。如果$(Z_1,...,Z_N)$是一个可交换的序列的$N$随机变量的值在一些波兰空间$X$，我们表明，法律$μ_k$的第一$k$组件有一个表示的形式$μ_k=\int_{{\mathcal P}_{\frac{1}{N}}(X)} F_{N,k}(λ) \, \mbox{d} α(λ)$的一些概率措施$α$的集合上的$1/N$量化概率措施$X$和某些通用多项式$F_{N,k}$。后者包括一个领先的长期$N^{k-1}\! /{\small \prod_{j=1}^{k-1}(N\! -\! j)\, λ^{\otimes k}}$和一个有限的，指数衰减的一系列的相关修正的顺序$N^{-j}$（$j=1,...,k$）。该$F_{N,k}(λ)$正是极值这样的法律，表示通过一个明确的多项式公式在其单点边际$λ$。应用包括新的近似MMOT通过多项式凸化和识别的其余部分估计在著名的误差界Diaconis-Freedman之间的有限和无限的可交换的法律。

英文摘要：

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 $μ_k$ of the first $k$ components has a representation of the form $μ_k=\int_{{\mathcal P}_{\frac{1}{N}}(X)} F_{N,k}(λ) \, \mbox{d} α(λ)$ 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 $N^{k-1}\! /{\small \prod_{j=1}^{k-1}(N\! -\! j)\, λ^{\otimes k}}$ 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-Freedman between finite and infinite exchangeable laws.

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