In applications of Bayesian procedures, even when the prior law is carefully specified, it may be delicate to elicit the prior hyperparameters so that it is often tempting to fix them from the data, usually by their maximum likelihood estimates (MMLE), obtaining a so-called empirical Bayes posterior distribution. Although questionable, this is a common practice; but theoretical properties seem mostly only available on a case-by-case basis. In this paper we provide general properties for parametric models. First, we study the limit behavior of the MMLE and prove results in quite general settings, while also conceptualizing the frequentist context as an unexplored case of maximum likelihood estimation under model misspecification. We cover both identifiable models, illustrating applications to sparse regression, and non-identifiable models - specifically, overfitted mixture models. Finally, we prove higher order merging results. In regular cases, the empirical Bayes posterior is shown to be a fast approximation to the Bayesian posterior distribution of the researcher who, within the given class of priors, has the most information about the true model's parameters. This is a faster approximation than classic Bernstein-von Mises results. Given the class of priors, our work provides formal contents to common beliefs on this popular practice.

翻译：在贝叶斯方法的实际应用中，即使先验分布被精心设定，先验超参数往往难以通过专家启发确定，因此研究者常倾向于通过数据估计这些参数（通常采用最大边际似然估计，MMLE），从而得到所谓的经验贝叶斯后验分布。尽管这种做法存在争议，却是一种常见实践，但其理论性质通常仅能以个案方式证明。本文针对参数模型提供了通用性质。首先，我们研究MMLE的极限行为，并在相当一般的设定下证明相关结论，同时将频率主义框架概念化为模型误设下最大似然估计的一个未探索特例。我们覆盖了可识别模型（以稀疏回归为例阐明应用）与不可识别模型（尤其是过拟合混合模型）。最后，我们证明高阶合并结果。在正则情形下，经验贝叶斯后验被证明是对贝叶斯后验分布的快速近似——该贝叶斯后验对应于在给定先验类中掌握真实模型参数最多信息的研究者所对应的后验分布。这种近似的收敛速度优于经典伯恩斯坦-冯·米塞斯定理所给出的结果。对于给定的先验类，我们的工作为这种流行实践中的常见认知提供了形式化理论支撑。