Bayesian inverse problems with Gaussian priors

Details Bayesian inverse problems with Gaussian priors Bayesian inverse problems with Gaussian priors

2011-03-14

arxiv.org/abs/1103.2692v2

Annals of Statistics 2011, Vol. 39, No. 5, 2626-2657

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/11-AOS920 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

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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