The Annals of Statistics

Bayesian inverse problems with Gaussian priors

B. T. Knapik, A. W. van der Vaart, and J. H. van Zanten

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Abstract

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.

Article information

Source
Ann. Statist., Volume 39, Number 5 (2011), 2626-2657.

Dates
First available in Project Euclid: 22 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1324563350

Digital Object Identifier
doi:10.1214/11-AOS920

Mathematical Reviews number (MathSciNet)
MR2906881

Zentralblatt MATH identifier
1232.62079

Subjects
Primary: 62G05: Estimation 62G15: Tolerance and confidence regions
Secondary: 62G20: Asymptotic properties

Keywords
Credible set posterior distribution Gaussian prior rate of contraction

Citation

Knapik, B. T.; van der Vaart, A. W.; van Zanten, J. H. Bayesian inverse problems with Gaussian priors. Ann. Statist. 39 (2011), no. 5, 2626--2657. doi:10.1214/11-AOS920. https://projecteuclid.org/euclid.aos/1324563350


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