  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 $\theta^\circ$ that
generates the data. While this establishes posterior consistency, however, the
concentration rate depends on both $\theta^\circ$ 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 $\theta^\circ$ 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 $\theta^\circ$ 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 $\theta^\circ$ nor on the given class. Moreover,
convergence of the fully data-driven Bayes estimator at the oracle or at the
minimax rate is established.
