Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions

Despite the ubiquity of the Gaussian process regression model, few\ntheoretical results are available that account for the fact that parameters of\nthe covariance kernel typically need to be estimated from the dataset. This\narticle provides one of the first theoretical analyses in the context of\nGaussian process regression with a noiseless dataset. Specifically, we consider\nthe scenario where the scale parameter of a Sobolev kernel (such as a\nMat\\'{e}rn kernel) is estimated by maximum likelihood. We show that the maximum\nlikelihood estimation of the scale parameter alone provides significant\nadaptation against misspecification of the Gaussian process model in the sense\nthat the model can become "slowly" overconfident at worst, regardless of the\ndifference between the smoothness of the data-generating function and that\nexpected by the model. The analysis is based on a combination of techniques\nfrom nonparametric regression and scattered data interpolation. Empirical\nresults are provided in support of the theoretical findings.\n