Convergence of Gaussian Process Regression with Estimated Hyper-parameters and Applications in Bayesian Inverse Problems

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Abstract

This work is concerned with the convergence of Gaussian process regression. A particular focus is on hierarchical Gaussian process regression, where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a-priori unknown, and are learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. We provide a convergence analysis that (i) holds for any continuous function f to be emulated; and (ii) shows that convergence of Gaussian process regression is unaffected by the additional learning of hyper-parameters from data, and is guaranteed in a wide range of scenarios. As the primary motivation for the work is the use of Gaussian process regression to approximate the data likelihood in Bayesian inverse problems, we provide a bound on the error introduced in the Bayesian posterior distribution in this context.
Original languageEnglish
Pages (from-to) 1310–1337
Number of pages29
JournalSIAM/ASA Journal on Uncertainty Quantification
Volume8
Issue number4
Early online date19 Oct 2020
DOIs
Publication statusE-pub ahead of print - 19 Oct 2020

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