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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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

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

First available in Project Euclid: 22 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Credible set posterior distribution Gaussian prior rate of contraction


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.

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