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Gaussian process (GP) regression is a Bayesian nonparametric method forregression and interpolation, offering a principled way of quantifying theuncertainties of predicted function values. For the quantified uncertainties tobe well-calibrated, however, the kernel of the GP prior has to be carefullyselected. In this paper, we theoretically compare two methods for choosing thekernel in GP regression: cross-validation and maximum likelihood estimation.Focusing on the scale-parameter estimation of a Brownian motion kernel in thenoiseless setting, we prove that cross-validation can yield asymptoticallywell-calibrated credible intervals for a broader class of ground-truthfunctions than maximum likelihood estimation, suggesting an advantage of theformer over the latter. Finally, motivated by the findings, we propose interiorcross validation, a procedure that adapts to an even broader class ofground-truth functions.

Comparing Scale Parameter Estimators for Gaussian Process Interpolation  with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum  Likelihood