  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.
