  We investigate the problem of constructing Bayesian credible sets that are
honest and adaptive for the L2-loss over a scale of Sobolev classes with
regularity ranging between [D; 2D], for some given D in the context of the
signal-in-white-noise model. We consider a scale of prior distributions indexed
by a regularity hyper-parameter and choose the hyper-parameter both by marginal
likelihood empirical Bayes and by hierarchical Bayes method, respectively. Next
we consider a ball centered around the corresponding posterior mean with
prescribed posterior probability. We show by theory and examples that both the
empirical Bayes and the hierarchical Bayes credible sets give misleading,
overconfident uncertainty quantification for certain oddly behaving truth. Then
we construct a new empirical Bayes method based on risk estimation, which
provides the correct uncertainty quantification and optimal size.
