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In an indirect Gaussian sequence space model lower and upper bounds are derived for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value θ^0 that generates the data. While this establishes posterior consistency, however, the concentration rate depends on both θ^0 and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each θ^0 separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive. This means that, given a parameter θ^0 or a class of parameters, respectively, the posterior distribution contracts at the oracle rate or at the minimax rate over the class. Notably, the hierarchical prior does not depend neither on θ^0 nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.

Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models