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We investigate two empirical Bayes methods and a hierarchical Bayes methodfor adapting the scale of a Gaussian process prior in a nonparametricregression model. We show that all methods lead to a posterior contraction ratethat adapts to the smoothness of the true regression function. Furthermore, weshow that the corresponding credible sets cover the true regression functionwhenever this function satisfies a certain extrapolation condition. Thiscondition depends on the specific method, but is implied by a condition ofself-similarity. The latter condition is shown to be satisfied with probabilityone under the prior distribution.

Adaptive Bayesian credible sets in regression with a Gaussian process prior