  We investigate the frequentist coverage properties of Bayesian credible sets
in a general, adaptive, nonparametric framework. It is well known that the
construction of adaptive and honest confidence sets is not possible in general.
To overcome this problem we introduce an extra assumption on the functional
parameters, the so called "general polished tail" condition. We then show that
under standard assumptions both the hierarchical and empirical Bayes methods
results in honest confidence sets for sieve type of priors in general settings
and we characterize their size. We apply the derived abstract results to
various examples, including the nonparametric regression model, density
estimation using exponential families of priors, density estimation using
histogram priors and nonparametric classification model, for which we show that
their size is near minimax adaptive with respect to the considered specific
semi-metrics.
