Uncertainty quantification is crucial in statistical sciences. In statistics, uncertainty about an estimator is described with the help of confidence sets. But the construction of confidence sets can be challenging, especially in complex, nonparametric problems. A popular aspect of the Bayesian approach is the way of quantifying uncertainty with the help of sets with prescribed high posterior probability (called credible sets). In parametric models credible sets are asymptotically confidence sets as well. However, in nonparametric and high-dimensional models the question is still unanswered about how much one can trust Bayesian credible sets as a measure of confidence in the statistical procedure from a frequentist perspective (and a general approach for understanding the coverage of credible sets is missing, too). The paper under review aims to partially fill the gap by contributing to the fundamental understanding of this field. The authors derive results for general choices of models and sieve type of priors, in the spirit of [S. Ghosal et al., ibid. 28, No. 2, 500–531 (2000; Zbl 1105.62315); S. Ghosal and A. van der Vaart, ibid. 35, No. 1, 192–223 (2007; Zbl 1114.62060); the authors, ibid. 45, No. 2, 833–865 (2017; Zbl 1371.62048)]. Namely, the frequentist properties of Bayesian credible sets resulting from the hierarchical and the empirical Bayes procedures are investigated. The authors introduce conditions under which credible sets have honest frequentist coverage and rate adaptive size in the considered general setting. The derived results are then applied to various choices of sampling and prior models.