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arXiv 2101.05079math.PR
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用于多元连续分布的 Stein 密度方法

Stein's density method for multivariate continuous distributions

Guillaume Mijoule, Martin Raič, Gesine Reinert, Yvik Swan

备注：
This is an updated version of our paper arXiv:1806.03478 with some material left out. We leave paper arXiv:1806.03478 up on arXiv for future reference
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AI总结：

本文构建多元连续分布的 Stein 密度方法一般框架，提出不依赖归一化常数的算子、Stein 核及恒等式，并用于推导 Wasserstein 距离界与比较多种分布和 copula。

摘要：

本文为多元连续分布的 Stein 密度方法提供了一个一般框架。该方法将任意概率密度函数与一个典范算子和 Stein 类，以及无穷多个算子和类（我们称之为标准化）联系起来。这些又为任意定义在 $\mathbb{R}^d$ 上的连续分布生成了一整族 Stein 恒等式和刻画，其中我们重点介绍基于得分函数和 Stein 核的那些。这些算子的一个特点是它们不依赖于归一化常数。本文引入并考察了 Stein 核的一个新定义；通过与质量输运的联系得到了积分公式，同时也得到了椭圆分布的现成可用的显式公式。利用这些核的灵活性，我们在 Stein 差异（并因此在 2-Wasserstein 距离）中比较了两个正态分布、Student 分布与正态分布，以及两个 normal-gamma 分布。本文给出了连续分布之间 1-Wasserstein 距离的上界和下界，并针对多种例子进行了计算：不同正态分布之间的比较（在某些情形下改进了已有界）、Bayes 设定中具有不同先验的后验分布（包括 logistic 回归）、中心化的 Azzalini–Dalla Valle 分布。最后，本文引入了弱 Stein 方程和弱 Stein 因子的概念。若分布具有 Poincaré 常数，则对于 Lipschitz 测试函数可得到弱 Stein 方程解的界。我们利用这些界在 1-Wasserstein 距离下比较单位正方形上的不同 copula。

英文摘要：

This paper provides a general framework for Stein's density method for multivariate continuous distributions. The approach associates to any probability density function a canonical operator and Stein class, as well as an infinite collection of operators and classes which we call standardizations. These in turn spawn an entire family of Stein identities and characterizations for any continuous distribution on $\mathbb{R}^d$, among which we highlight those based on the score function and the Stein kernel. A feature of these operators is that they do not depend on normalizing constants. A new definition of Stein kernel is introduced and examined; integral formulas are obtained through a connection with mass transport, as well as ready-to-use explicit formulas for elliptical distributions. The flexibility of the kernels is used to compare in Stein discrepancy (and therefore 2-Wasserstein distance) between two normal distributions, Student and normal distributions, as well as two normal-gamma distributions. Upper and lower bounds on the 1-Wasserstein distance between continuous distributions are provided, and computed for a variety of examples: comparison between different normal distributions (improving on existing bounds in some regimes), posterior distributions with different priors in a Bayesian setting (including logistic regression), centred Azzalini--Dalla Valle distributions. Finally the notion of weak Stein equation and weak Stein factors is introduced. Bounds for solutions of the weak Stein equation are obtained for Lipschitz test functions if the distribution admits a Poincaré constant. We use these bounds to compare different copulas on the unit square in 1-Wasserstein distance.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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