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October 2011 Bayesian inverse problems with Gaussian priors
B. T. Knapik, A. W. van der Vaart, J. H. van Zanten
Ann. Statist. 39(5): 2626-2657 (October 2011). DOI: 10.1214/11-AOS920


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.


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B. T. Knapik. A. W. van der Vaart. J. H. van Zanten. "Bayesian inverse problems with Gaussian priors." Ann. Statist. 39 (5) 2626 - 2657, October 2011.


Published: October 2011
First available in Project Euclid: 22 December 2011

zbMATH: 1232.62079
MathSciNet: MR2906881
Digital Object Identifier: 10.1214/11-AOS920

Primary: 62G05 , 62G15
Secondary: 62G20

Keywords: Credible set , Gaussian prior , posterior distribution , rate of contraction

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 5 • October 2011
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