Bayesian inverse problems with Gaussian priors
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.
 Publication:

arXiv eprints
 Pub Date:
 March 2011
 arXiv:
 arXiv:1103.2692
 Bibcode:
 2011arXiv1103.2692K
 Keywords:

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