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

被引：32

作者：

Knapik, B. T. ^{[1
]}

Szabo, B. T. ^{[2
]}

van der Vaart, A. W. ^{[3
]}

van Zanten, J. H. ^{[4
]}

机构：

[1] Vrije Univ Amsterdam, Dept Math, Amsterdam, Netherlands

[2] Budapest Univ Technol & Econ, Dept Stochast, Budapest, Hungary

[3] Leiden Univ, Math Inst, Leiden, Netherlands

[4] Univ Amsterdam, Korteweg de Vries Inst Math, Amsterdam, Netherlands

来源：

PROBABILITY THEORY AND RELATED FIELDS | 2016年 / 164卷 / 3-4期

基金：

欧洲研究理事会;

关键词：

Adaptation; Empirical Bayes; Hierarchical Bayes; Posterior distribution; Gaussian prior; Rate of convergence; Nonparametric inverse problems; EMPIRICAL BAYES; CONVERGENCE-RATES;

D O I：

10.1007/s00440-015-0619-7

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

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.

引用

页码：771 / 813

页数：43

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