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.Comment: Published in at http://dx.doi.org/10.1214/11-AOS920 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies regularity. We prove that both methods we consider succeed in automatically selecting this parameter optimally, resulting in optimal convergence rates for truths with Sobolev or analytic "smoothness", without using knowledge about this regularity. Both methods are illustrated by simulation examples.
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