Measuring dependence in the Wasserstein distance for Bayesian nonparametric models
用 Wasserstein 距离度量贝叶斯非参数模型的依赖性AI 解读 ↓
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用 Wasserstein 距离度量贝叶斯非参数模型的依赖性
- 研究问题
- 随机测度构造的贝叶斯非参数模型中，如何建立不依赖具体模型的框架，对随机概率测度之间的依赖结构进行量化？
- 方法与思路
- 作者以同边际共单调耦合为参照，用 Wasserstein 距离衡量接近程度，并借助复合 Poisson 近似推导依赖度量的界。
- 材料陈述的贡献
- 摘要称提出一般性依赖量化框架，得到以底层 Lévy 强度表示的界，并将结果用于若干常见贝叶斯模型的分析。
- 阅读关注点
- 应核对交换性与共单调参照的精确定义、距离如何转化为依赖指标、边际固定的要求，以及近似和界的适用条件。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；摘要未说明指标的完整性质、界的精度或模型实例，不能仅凭 Wasserstein 一词判作分布效应。
原文摘要
The proposal and study of dependent Bayesian nonparametric models has been one of the most active research lines in the last two decades, with random vectors of measures representing a natural and popular tool to define them. Nonetheless, a principled approach to understand and quantify the associated dependence structure is still missing. We devise a general, and not model-specific, framework to achieve this task for random measure based models, which consists in: (a) quantify dependence of a random vector of probabilities in terms of closeness to exchangeability, which corresponds to the maximally dependent coupling with the same marginal distributions, that is, the comonotonic vector; (b) recast the problem in terms of the underlying random measures (in the same Fréchet class) and quantify the closeness to comonotonicity; (c) define a distance based on the Wasserstein metric, which is ideally suited for spaces of measures, to measure the dependence in a principled way. Several results, which represent the very first in the area, are obtained. In particular, useful bounds in terms of the underlying Lévy intensities are derived relying on compound Poisson approximations. These are then specialized to popular models in the Bayesian literature leading to interesting insights.