论文标题
通过Wasserstein Dynamics进行后验收率的新方法
A new approach to posterior contraction rates via Wasserstein dynamics
论文作者
论文摘要
本文为贝叶斯统计中量化后验收率（PCR）的经典问题提供了一种新的方法。我们的方法依赖于瓦斯汀的距离，这导致了两种主要贡献，这些贡献改善了现有的PCR文献。第一个贡献利用了Wasserstein距离的动态公式，用于简短称为Wasserstein Dynamics，以便在主导的贝叶斯统计模型下建立PCR。 As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both stating and proving PCRs, and it sets forth a natural connection between PCRs and three well-known classical problems in statistics and probability theory: the speed of mean Glivenko-Cantelli convergence, the estimation of weighted Poincaré-Wirtinger constants and Sanov large deviation principle for Wasserstein distance.第二个贡献结合了Wasserstein距离的使用与合适的筛子结构，以在完整的贝叶斯非参数模型下建立PCR。作为对现有PCR文献的新颖性，我们的第二个结果为非主导贝叶斯模型的PCR进行了首次处理。我们的结果应用于某些经典的贝叶斯统计模型，例如常规参数模型，无限维指数族家族，无限维度的线性回归以及在dirichlet过程先验下的非参数模型。
This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. The first contribution exploits the dynamic formulation of Wasserstein distance, for short referred to as Wasserstein dynamics, in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both stating and proving PCRs, and it sets forth a natural connection between PCRs and three well-known classical problems in statistics and probability theory: the speed of mean Glivenko-Cantelli convergence, the estimation of weighted Poincaré-Wirtinger constants and Sanov large deviation principle for Wasserstein distance. The second contribution combines the use of Wasserstein distance with a suitable sieve construction to establish PCRs under full Bayesian nonparametric models. As a novelty with respect to existing literature of PCRs, our second result provides with the first treatment of PCRs under non-dominated Bayesian models. Applications of our results are presented for some classical Bayesian statistical models, e.g., regular parametric models, infinite-dimensional exponential families, linear regression in infinite dimension and nonparametric models under Dirichlet process priors.
