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.
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