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Abstract

Considering a family of statistical, linear, ill-posed inverse problems, we propose their study from two perspectives, the Bayesian and frequentist paradigms. Under the Bayesian paradigm, we investigate two different asymptotic analyses for Gaus- sian sieve priors and their hierarchical counterpart. The first analysis is with respect to an iteration procedure, where the posterior distribution is used as a prior to compute a new posterior distribution while using the same likelihood and data. We are interested in the limit of the sequence of distributions generated this way, if it exists. The second analysis, more traditionally, investigates the behaviour of the posterior distri- bution as the amount of data increases. Assuming the existence of a true parameter, one is then interested in showing that the posterior distribution contracts around the truth at an optimal rate. We illustrate all those results by their application to the inverse Gaussian sequence space model. Finally we exhibit that the posterior mean of the hierarchical Gaussian sieve prior is both a shrinkage and an aggregation estimator, with interesting optimality properties. Motivated by the last findings about posterior mean of hierarchical Gaussian sieves, we propose to investigate the quadratic risk of aggregation estimators, which shape mimics the one of the above-mentioned posterior means. We introduce a strategy, relying on the decomposition of the risk, which allows to obtain optimal convergence rates in the cases of known and unknown operator, for dependent as well as absolutely regular data. We demonstrate the use of this method on the inverse Gaussian sequence space model as well as the circular density deconvolution and obtained optimality results under mild hypotheses.