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 covariance 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.

翻译：高斯过程（GP）回归是一种用于回归与插值的贝叶斯非参数方法，能够以规范方式量化预测函数值的不确定性。然而，为使量化后的不确定性得到良好校准，必须谨慎选择GP先验的协方差核。本文从理论上比较了GP回归中两种核选择方法：交叉验证与最大似然估计。聚焦于无噪声设定下布朗运动核的尺度参数估计，我们证明：相较于最大似然估计，交叉验证能在更广泛的地面真值函数类别上产生渐近校准良好的可信区间，这表明前者相较于后者具有优势。