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

Toni Karvonen, George Wynne, Filip Tronarp, Chris Oates, Simo Särkkä

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)
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Abstract

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 Matern 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.

Original languageEnglish
Pages (from-to)926-958
Number of pages33
JournalSIAM/ASA Journal on Uncertainty Quantification
Volume8
Issue number3
DOIs
Publication statusPublished - 2020
MoE publication typeA1 Journal article-refereed

Keywords

  • Bayesian cubature
  • Credible sets
  • Model misspecification
  • Nonparametric regression
  • Scattered data approximation

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