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