Abstract
Bayesian inference and uncertainty quantification in a general class of nonlinear inverse regression models is considered. Analytic conditions on the regression model and on Gaussian process priors for θ are provided such that semiparametrically efficient inference is possible for a large class of linear functionals of θ. A general Bernstein–von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence, posterior-based credible sets are valid and optimal from a frequentist point of view. The theory is illustrated with two applications with PDEs that arise in nonlinear tomography problems: an elliptic inverse problem for a Schrödinger equation, and inversion of non-Abelian X-ray transforms. New analytical techniques are deployed to show that the relevant Fisher information operators are invertible between suitable function spaces.
Funding Statement
F. Monard was supported by NSF Grant DMS-1814104 and NSF CAREER Grant DMS-1943580. R. Nickl was supported by the European Research Council under ERC Grant 647812 (UQMSI). G.P. Paternain was supported by the Leverhulme trust and EPSRC Grant EP/R001898/1.
Acknowledgments
The authors would like to thank the anonymous referees for their constructive comments that improved the quality of this paper.
Funding Statement
F. Monard was supported by NSF Grant DMS-1814104 and NSF CAREER Grant DMS-1943580. R. Nickl was supported by the European Research Council under ERC Grant 647812 (UQMSI). G.P. Paternain was supported by the Leverhulme trust and EPSRC Grant EP/R001898/1.
Acknowledgments
The authors would like to thank the anonymous referees for their constructive comments that improved the quality of this paper.
Citation
François Monard. Richard Nickl. Gabriel P. Paternain. "Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors." Ann. Statist. 49 (6) 3255 - 3298, December 2021. https://doi.org/10.1214/21-AOS2082
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