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The paper [, we obtain an adaptive procedure for the inverse problem using the hierarchical prior.Optimal convergence rates are obtained (up to lower order factors), uniformly over Sobolev and analytic scales. This empirical Bayes procedure is not really Bayesian in the strict sense of the word.These methods are generally considered to be preferable to methods that use only a single, fixed value of the hyper-parameters.In the inverse problem setting it is known from the recent paper [] that using a fixed prior can indeed be undesirable, since it can lead to convergence rates that are sub-optimal, unless by chance the statistician has selected a prior that captures the fine properties of the unknown parameter (like its degree of smoothness, if it is a function).Theoretical work that supports the preference for empirical or hierarchical Bayes methods does not exist at the present time however.It has until now been unknown whether these approaches can indeed robustify a procedure against prior mismatch.

In recent years, Bayesian approaches have become more and more common in dealing with nonparametric statistical inverse problems.One possible explanation of the increasing popularity of Bayesian methods is the fact that assigning a prior distribution to an unknown functional parameter is a natural way of specifying a degree of regularization.Probably at least as important is the fact that various computational methods exist to carry out the inference in practice, including MCMC methods and approximate methods like expectation propagation, Laplace approximations and approximate Bayesian computation.This development parallels the earlier work on adaptive methods for the direct nonparametric problem to some extent, although the inverse case is technically usually more demanding.The adaptive methods typically involve a data-driven choice of a tuning parameter in order to automatically achieve an optimal bias-variance trade-off, as in Lepski’s method for instance.

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