Choosing observation operators to mitigate model error in Bayesian inverse problems

TL;DR

This paper develops criteria for selecting observation operators in Bayesian inverse problems to reduce the impact of model error, using stability estimates and KL divergence bounds, demonstrated on a PDE inverse problem.

Contribution

It introduces a novel approach to choose observation operators that mitigate model error effects in Bayesian inverse problems using stability bounds.

Findings

Derived bounds for posterior KL divergence under likelihood perturbations.

Provided criteria for observation operator selection to reduce model error impact.

Validated the approach on an advection-diffusion-reaction PDE inverse problem.

Abstract

In statistical inference, a discrepancy between the parameter-to-observable map that generates the data and the parameter-to-observable map that is used for inference can lead to misspecified likelihoods and thus to incorrect estimates. In many inverse problems, the parameter-to-observable map is the composition of a linear state-to-observable map called an `observation operator' and a possibly nonlinear parameter-to-state map called the `model'. We consider such Bayesian inverse problems where the discrepancy in the parameter-to-observable map is due to the use of an approximate model that differs from the best model, i.e. to nonzero `model error'. Multiple approaches have been proposed to address such discrepancies, each leading to a specific posterior. We show how to use local Lipschitz stability estimates of posteriors with respect to likelihood perturbations to bound the Kullback--Leibler divergence of the posterior of each approach with respect to the posterior associated to the best model. Our bounds lead to criteria for choosing observation operators that mitigate the effect of model error for Bayesian inverse problems of this type. We illustrate one such criterion on an advection-diffusion-reaction PDE inverse problem from the literature, and use this example to discuss the importance and challenges of model error-aware inference.