Adaptive Bayesian estimation in indirect Gaussian sequence space models

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.

Reference:
Johannes, J., Simoni, A. and Schenk R. (2015). Adaptive Bayesian estimation in indirect Gaussian sequence space models (arXiv:1502.00184)