  We consider the problem of constructing Bayesian based confidence sets for
linear functionals in the inverse Gaussian white noise model. We work with a
scale of Gaussian priors indexed by a regularity hyper-parameter and apply the
data-driven (slightly modified) marginal likelihood empirical Bayes method for
the choice of this hyper-parameter. We show by theory and simulations that the
credible sets constructed by this method have sub-optimal behaviour in general.
However, by assuming "self-similarity" the credible sets have rate-adaptive
size and optimal coverage. As an application of these results we construct
$L_{\infty}$-credible bands for the true functional parameter with adaptive
size and optimal coverage under self-similarity constraint.
