Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

TL;DR

This paper derives non-asymptotic error bounds for the Laplace approximation in nonlinear Bayesian inverse problems, accounting for nonlinearity and problem dimension, and shows the approximation error is proportional to the perturbation size.

Contribution

It provides novel, explicit error estimates for the Laplace approximation in nonlinear inverse problems, considering nonlinearity and dimension effects.

Findings

Laplace approximation error scales with the nonlinearity perturbation

Error estimates are valid for fixed noise levels

Insights into linearization accuracy in Bayesian inference

Abstract

In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (2020), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.