Laplace priors and spatial inhomogeneity in Bayesian inverse problems

Sven Wang

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 \textit{\ser{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 assumptions on the forward map. We then verify the assumptions for two non-linear inverse problems arising from elliptic partial differential equations, the \textit{Darcy flow} model from geophysics as well as a model for the \textit{Schr\"odinger equation} appearing in tomography. In the course of the proofs, we also obtain novel concentration inequalities for penalized least squares estimators with $\ell^1$ 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 $B^{\alpha}_{11}$, $\alpha>0$.

In a setting with direct observations, we complement these upper bounds with a lower bound on the rate of contraction for \textit{arbitrary} Gaussian priors. An immediate consequence of our results is that while Laplace priors are able to achieve minimax-optimal rates over $B^{\alpha}_{11}$-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 indeed compatible with $\ell^1$ regularity structure in the underlying parameter.