May 2024 Laplace priors and spatial inhomogeneity in Bayesian inverse problems
Sergios Agapiou, Sven Wang
Author Affiliations +
Bernoulli 30(2): 878-910 (May 2024). DOI: 10.3150/22-BEJ1563

Abstract

Spatially inhomogeneous functions, which may be smooth in some regions and rough in other regions, are modelled naturally in a Bayesian manner using so-called Besov priors which are given by random wavelet expansions with Laplace-distributed coefficients. This paper studies theoretical guarantees for such prior measures – specifically, we examine their frequentist posterior contraction rates in the setting of non-linear inverse problems with Gaussian white noise. Our results are first derived under a general local Lipschitz assumption on the forward map. We then verify the assumption for two non-linear inverse problems arising from elliptic partial differential equations, the Darcy flow model from geophysics as well as a model for the Schrödinger equation appearing in tomography. In the course of the proofs, we also obtain novel concentration inequalities for penalized least squares estimators with 1 wavelet penalty, which have a natural interpretation as maximum a posteriori (MAP) estimators. The true parameter is assumed to belong to some spatially inhomogeneous Besov class B11α, with α>0 sufficiently large. In a setting with direct observations, we complement these upper bounds with a lower bound on the rate of contraction for arbitrary Gaussian priors. An immediate consequence of our results is that while Laplace priors can achieve minimax-optimal rates over B11α-classes, Gaussian priors are limited to a (by a polynomial factor) slower contraction rate. This gives information-theoretical justification for the intuition that Laplace priors are more compatible with 1 regularity structure in the underlying parameter.

Funding Statement

SW gratefully acknowledges financial support by the European Research Council, ERC grant agreement 647812 (UQMSI) as well as (during the finishing stages of this work) the Air Force Office of Scientific Research under award number FA9550-20-1-0397.

Acknowledgements

The authors would like to thank Richard Nickl for suggesting this collaboration and for funding SA’s research visit to Cambridge during which this work was initiated. We thank three anonymous referees, the AE and the editor for several helpful remarks.

Citation

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Sergios Agapiou. Sven Wang. "Laplace priors and spatial inhomogeneity in Bayesian inverse problems." Bernoulli 30 (2) 878 - 910, May 2024. https://doi.org/10.3150/22-BEJ1563

Information

Received: 1 March 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699538
Digital Object Identifier: 10.3150/22-BEJ1563

Keywords: Bayesian nonparametric inference , frequentist consistency , Inverse problems , Laplace prior , spatially inhomogeneous functions

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Vol.30 • No. 2 • May 2024
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