Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the data set. This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless data set. Specifically, we consider the scenario where the scale parameter of a Sobolev kernel (such as a Matérn kernel) is estimated by maximum likelihood. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model in the sense that the model can become “slowly” overconfident at worst, regardless of the difference between the smoothness of the data-generating function and that expected by the model. The analysis is based on a combination of techniques from nonparametric regression and scattered data interpolation. Empirical results are provided in support of the theoretical findings.


  1. nonparametric regression
  2. scattered data approximation
  3. credible sets
  4. Bayesian cubature
  5. model misspecification

MSC codes

  1. 60G15
  2. 62G20
  3. 68T37
  4. 65D05
  5. 46E22

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Information & Authors


Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 926 - 958
ISSN (online): 2166-2525


Submitted: 30 January 2020
Accepted: 12 May 2020
Published online: 4 August 2020


  1. nonparametric regression
  2. scattered data approximation
  3. credible sets
  4. Bayesian cubature
  5. model misspecification

MSC codes

  1. 60G15
  2. 62G20
  3. 68T37
  4. 65D05
  5. 46E22



Funding Information

Aalto ELEC Doctoral School
Academy of Finland https://doi.org/10.13039/501100002341
Alan Turing Institute https://doi.org/10.13039/100012338
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : 18000171

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