Laplace's method in Bayesian inverse problems

TL;DR

This paper develops an explicit Laplace approximation for Bayesian inverse problems in infinite-dimensional spaces, providing bounds on the approximation error using the Hellinger distance.

Contribution

It derives a second-order Gaussian approximation of the posterior measure and explicitly bounds the error in infinite-dimensional settings.

Findings

Explicit Laplace approximation in infinite dimensions

Bound on Hellinger distance between true and approximate posterior

Facilitates efficient sampling in Bayesian inverse problems

Abstract

In a Bayesian inverse problem setting, the solution consists of a posterior measure obtained by combining prior belief, information about the forward operator, and noisy observational data. This measure is most often given in terms of a density with respect to a reference measure in a high-dimensional (or infinite-dimensional) Banach space. Although Monte Carlo sampling methods provide a way of querying the posterior, the necessity of evaluating the forward operator many times (which will often be a costly PDE solver) prohibits this in practice. For this reason, many practitioners choose a suitable Gaussian approximation of the posterior measure, in a procedure called Laplace's method. Once generated, this Gaussian measure is a lot easier to sample from and properties like moments are immediately acquired. This paper derives Laplace's approximation of the posterior measure attributed to the inverse problem explicitly as the posterior measure of a second-order approximation of the data-misfit functional, specifically in the infinite-dimensional setting. By use of a reverse Cauchy-Schwarz inequality we are able to explicitly bound the Hellinger distance between the posterior and its approximation.