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We study empirical and hierarchical Bayes approaches to the problem ofestimating an infinite-dimensional parameter in mildly ill-posed inverseproblems. We consider a class of prior distributions indexed by ahyperparameter that quantifies regularity. We prove that both methods weconsider succeed in automatically selecting this parameter optimally, resultingin optimal convergence rates for truths with Sobolev or analytic "smoothness",without using knowledge about this regularity. Both methods are illustrated bysimulation examples.

Bayes procedures for adaptive inference in inverse problems for the white noise model