BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS

• 成果类型: Article
• 署名作者: Knapik, B. T.; van der Vaart, A. W.; van Zanten, J. H.
• 署名单位: Vrije Universiteit Amsterdam; Eindhoven University of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS920
• 发表日期: 2011
• 页码: 2626-2657
• 关键词: posterior distributions; Asymptotic Normality; convergence-rates; hilbert scales; regularization; functionals; parameters; inference

摘要

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage. In the latter case credible sets are of the correct order of magnitude. The results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive.