Laplace priors and spatial inhomogeneity in Bayesian inverse problems
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Author(s)
Agapiou, Sergios
E N Wa Ng, S
Date Issued
May 2024
Journal
Bernoulli
Citation
Sergios Agapiou. Sven Wang. "Laplace priors and spatial inhomogeneity in Bayesian inverse problems." Bernoulli 30 (2) 878 - 910, May 2024.
Version
Final published version
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 B α 11 , 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 B α 11-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.
Subjects
Bayesian nonparametric inference
frequentist consistency
inverse problems
Laplace prior
spatially inhomogeneous functions
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.3150/22-BEJ1563
