2011/03/31 by B. T. Knapik, Aad van der Vaart, A. W. van der Vaart +1 · 3 citations
Computer Science · Mathematics · #Applied mathematics #Bayesian inference #Bayesian probability #Frequentist inference #Gaussian Processes and Bayesian Inference #Inverse problem #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematical optimization #Mathematics #Minimax #Posterior probability #Prior probability #Scale parameter #Smoothness #Statistical Methods and Inference #Statistics #math.ST #stat.TH
paper · pdf · doi:10.1214/11-aos920
published as Annals of Statistics 2011, Vol. 39, No. 5, 2626-2657 · Published in at http://dx.doi.org/10.1214/11-AOS920 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2011/10/01 · arxiv created 2012/02/23 · arxiv updated 2012/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.