Comparing scale parameter estimators for Gaussian process interpolation with the brownian motion prior: Leave-one-out cross validation and maximum likelihood

Gaussian process (GP) regression is a Bayesian nonparametric method for regression and interpolation that offers 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 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.