  For a general statistical model, we introduce the notion of data dependent
measure (DDM) on the model parameter. Typical examples of DDM are the posterior
distributions. Like for posteriors, the quality of a DDM is characterized by
the contraction rate which we allow to be local, i.e., depending on the
parameter. We construct confidence sets as DDM-credible sets and address the
issue of optimality of such sets, via a trade-off between its "size" (the local
radial rate) and its coverage probability. In the mildly ill-posed inverse
signal-in-white-noise model, we construct a DDM as empirical Bayes posterior
with respect to a certain prior, and define its (default) credible set. Then we
introduce 'excessive bias restriction' (EBR), more general than
'self-similarity' and 'polished tail condition' recently studied in the
literature. Under EBR, we establish the confidence optimality of our credible
set with some local (oracle) radial rate. We also derive the oracle estimation
inequality and the oracle DDM-contraction rate, non-asymptotically and
uniformly in $\ell_2$. The obtained local results are more powerful than
global: adaptive minimax results for a number of smoothness scales follow as
consequence, in particular, the ones considered by Szabo, van der Vaart and van
Zanten (2015).
