In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar\'e-Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincar\'e-Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.

翻译：在贝叶斯统计学中，后验收缩速率（Posterior Contraction Rates, PCRs）量化了随着样本量趋于无穷时，后验分布以适当方式向真实模型任意小邻域收敛的速度。本文针对函数参数空间上的强范数距离，提出了一种研究后验收缩速率的新方法。该方法的核心在于将后验分布的局部Lipschitz连续性条件与Wasserstein距离的动态表述相结合，从而揭示了后验收缩速率与数学分析、概率论和统计学中若干经典问题之间的有趣联系，例如：用于近似积分的Laplace方法、Wasserstein距离下的Sanov大偏差原理、均值Glivenko-Cantelli定理的收敛速率以及加权Poincaré-Wirtinger常数的估计。我们首先针对正则无穷维指数族模型给出了一个关于后验收缩速率的定理，该定理利用了模型的充分统计量，随后将此定理推广至一般控制模型。这些结果依赖于我们发展的用于评估无穷维Laplace积分和加权Poincaré-Wirtinger常数的新技术，这些技术本身也具有独立研究价值。所提出的方法被应用于正则参数模型、多项模型、有限维和无穷维Logistic-Gaussian模型以及无穷维线性回归。总体而言，我们的方法在有限维模型中导出了最优后验收缩速率，而对于无穷维模型，则明确展示了先验分布如何影响后验收缩速率。