Laplace priors and spatial inhomogeneity in Bayesian inverse problems

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 ℓ1 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 ℓ1 regularity structure in the underlying parameter.

Locations

• Bernoulli - View
• arXiv (Cornell University) - View - PDF