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Posterior consistency for Gaussian process approximations of Bayesian posterior distributions


Authors: Andrew M. Stuart and Aretha L. Teckentrup
Journal: Math. Comp. 87 (2018), 721-753
MSC (2010): Primary 60G15, 62G08, 65D05, 65D30, 65J22
DOI: https://doi.org/10.1090/mcom/3244
Published electronically: August 3, 2017
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Abstract: We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator. Our analysis includes approximations based on the mean of the predictive process, as well as approximations based on the full Gaussian process emulator. Our results show that the Hellinger distance between the true posterior and its approximations can be bounded by moments of the error in the emulator. Numerical results confirm our theoretical findings.


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Additional Information

Andrew M. Stuart
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, England
Address at time of publication: Computing and Mathematical Sciences, Caltech, Pasadena, California 91125
Email: astuart@caltech.edu

Aretha L. Teckentrup
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, England
Address at time of publication: School of Mathematics, James Clerk Maxwell Building, University of Edinburgh, EH9 3FD, Edinburgh, Scotland
Email: a.teckentrup@ed.ac.uk

DOI: https://doi.org/10.1090/mcom/3244
Keywords: Inverse problem, Bayesian approach, surrogate model, Gaussian process regression, posterior consistency
Received by editor(s): March 7, 2016
Received by editor(s) in revised form: September 26, 2016
Published electronically: August 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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