ASYMPTOTIC BEHAVIOUR OF THE EMPIRICAL BAYES POSTERIORS ASSOCIATED TO MAXIMUM MARGINAL LIKELIHOOD ESTIMATOR

• 成果类型: Article
• 署名作者: Rousseau, Judith; Szabo, Botond
• 署名单位: Universite PSL; Universite Paris-Dauphine; Institut Polytechnique de Paris; ENSAE Paris; Budapest University of Technology & Economics; Leiden University - Excl LUMC; Leiden University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1469
• 发表日期: 2017
• 页码: 833-865
• 关键词: dimensional exponential-families; Gaussian process priors; von mises theorem; Inverse problems; convergence-rates; credible sets; distributions; regression; contraction; functionals

摘要

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes,posterior distribution in general setting. First, we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.