标题： 贝叶斯估计量在一般损失函数下的弱收敛性 显示英文标题

标题： Weak convergence of Bayes estimators under general loss functions

摘要： 我们研究在超越经典平移不变设置的广义损失函数类下参数贝叶斯估计量的渐近行为。为此，我们开发了一个统一的理论框架，用于具有局部多项式结构的损失函数。这一一般理论涵盖了重要的例子，如平方Wasserstein距离、Sinkhorn散度和Stein差异，这些在现代统计推断和机器学习中得到了广泛关注。在经典Bernstein--von Mises定理的基础上，我们建立了贝叶斯估计量继承后验渐近正态性的充分条件。作为副产品，我们还推导了Wasserstein诱导的损失函数的可微性条件，并为贝叶斯估计量提供了新的相容性结果。多个例子和数值实验展示了所提出方法的相关性和准确性。 显示英文摘要

摘要： We investigate the asymptotic behavior of parametric Bayes estimators under a broad class of loss functions that extend beyond the classical translation-invariant setting. To this end, we develop a unified theoretical framework for loss functions exhibiting locally polynomial structure. This general theory encompasses important examples such as the squared Wasserstein distance, the Sinkhorn divergence and Stein discrepancies, which have gained prominence in modern statistical inference and machine learning. Building on the classical Bernstein--von Mises theorem, we establish sufficient conditions under which Bayes estimators inherit the posterior's asymptotic normality. As a by-product, we also derive conditions for the differentiability of Wasserstein-induced loss functions and provide new consistency results for Bayes estimators. Several examples and numerical experiments demonstrate the relevance and accuracy of the proposed methodology.