Bayes procedures for adaptive inference in inverse problems for the white noise model

• 成果类型: Article
• 署名作者: Knapik, B. T.; Szabo, B. T.; van der Vaart, A. W.; van Zanten, J. H.
• 署名单位: Vrije Universiteit Amsterdam; Budapest University of Technology & Economics; Leiden University - Excl LUMC; Leiden University; University of Amsterdam
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0619-7
• 发表日期: 2016
• 页码: 771-813
• 关键词: Empirical Bayes; convergence-rates

摘要

We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies regularity. We prove that both methods we consider succeed in automatically selecting this parameter optimally, resulting in optimal convergence rates for truths with Sobolev or analytic smoothness, without using knowledge about this regularity. Both methods are illustrated by simulation examples.