Bayes and empirical Bayes: do they merge?

• 成果类型: Article
• 署名作者: Petrone, S.; Rousseau, J.; Scricciolo, C.
• 署名单位: Bocconi University; Institut Polytechnique de Paris; ENSAE Paris
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/ast067
• 发表日期: 2014
• 页码: 285302
• 关键词: posterior distributions; nonparametric-inference; hierarchical Bayes; convergence-rates; priors; Consistency; mixtures; likelihood

摘要

Bayesian inference is attractive due to its internal coherence and for often having good frequentist properties. However, eliciting an honest prior may be difficult, and common practice is to take an empirical Bayes approach using an estimate of the prior hyperparameters. Although not rigorous, the underlying idea is that, for a sufficiently large sample size, empirical Bayes methods should lead to similar inferential answers as a proper Bayesian inference. However, precise mathematical results on this asymptotic agreement seem to be missing. In this paper, we give results in terms of merging Bayesian and empirical Bayes posterior distributions. We study two notions of merging: Bayesian weak merging and frequentist merging in total variation. We also show that, under regularity conditions, the empirical Bayes approach asymptotically gives an oracle selection of the prior hyperparameters. Examples include empirical Bayes density estimation with Dirichlet process mixtures.