We investigate two empirical Bayes methods and a hierarchical Bayes method for adapting the scale of a Gaussian process prior in a nonparametric regression model. We show that all methods lead to a posterior contraction rate that adapts to the smoothness of the true regression function. Furthermore, we show that the corresponding credible sets cover the true regression function whenever this function satisfies a certain extrapolation condition. This condition depends on the specific method, but is implied by a condition of self-similarity. The latter condition is shown to be satisfied with probability one under the prior distribution.
"Adaptive Bayesian credible sets in regression with a Gaussian process prior." Electron. J. Statist. 9 (2) 2475 - 2527, 2015. https://doi.org/10.1214/15-EJS1078