Posterior contraction under misspecification and heteroscedasticity in non-linear inverse problems

TL;DR

This paper establishes theoretical guarantees for surrogate posterior distributions in heteroscedastic nonlinear inverse problems, demonstrating their reliability and convergence rates even under model misspecification.

Contribution

It provides the first contraction results for surrogate posteriors in heteroscedastic nonlinear inverse problems with applications to PDE-constrained models.

Findings

Surrogate posteriors contract at rates comparable to exact posteriors.

The framework handles heteroscedastic noise and likelihood approximation errors.

Applications include reaction-diffusion and Navier-Stokes inverse problems.

Abstract

In many practical and numerical inverse problems, the exact data log-likelihood is not fully accessible, motivating the use of surrogate models. We study heteroscedastic nonparametric nonlinear regression problems with Gaussian errors and establish contraction results for posterior distributions arising from a surrogate log-likelihood constructed from proxy error variances, an approximate forward map, and an appropriate Gaussian process prior. Under general assumptions on the approximation quality, we show that the resulting surrogate posterior is statistically reliable and contracts about the true parameter at rates comparable to those of the exact posterior. The analysis leverages consistency properties of the (penalised) MLE to effectively handle heteroscedastic noise and to control the impact of likelihood approximation errors. We apply the framework to PDE-constrained inverse problems for a reaction-diffusion equation and the two-dimensional Navier-Stokes equation. In the latter case, we consider misspecified viscosity and forcing terms as well as Oseen-type linearization models, highlighting the relevance of our results for numerical analysis applications.