Bayesian aspects of some nonparametric problems

• 成果类型: Article
• 署名作者: Zhao, LH
• 署名单位: University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1016218229
• 发表日期: 2000
• 页码: 532-552
• 关键词: gaussian white-noise; asymptotic equivalence; regression; CONVERGENCE; priors; rates

摘要

We study the Bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation. We consider the asymptotic properties of Bayes procedures for conjugate (=Gaussian) priors. We show that so long as the prior puts nonzero measure on the very large parameter sat of interest then the Bayes estimators are not satisfactory. More specifically, we show that these estimators do not achieve the correct minim:ur rate over norm bounded sets in the parameter space. Thus all Bayes estimators for proper Gaussian priors have zero asymptotic efficiency in this minimax sense. We then present a class of priors whose Bayes procedures attain the optimal minimax I ate of convergence. These priors may be viewed as compound, or hierarchical, mixtures of suitable Gaussian distributions.