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In statistical inference, a discrepancy between the parameter-to-observablemap that generates the data and the parameter-to-observable map that is usedfor inference can lead to misspecified likelihoods and thus to incorrectestimates. In many inverse problems, the parameter-to-observable map is thecomposition of a linear state-to-observable map called an `observationoperator' and a possibly nonlinear parameter-to-state map called the `model'.We consider such Bayesian inverse problems where the discrepancy in theparameter-to-observable map is due to the use of an approximate model thatdiffers from the best model, i.e. to nonzero `model error'. Multiple approacheshave been proposed to address such discrepancies, each leading to a specificposterior. We show how to use local Lipschitz stability estimates of posteriorswith respect to likelihood perturbations to bound the Kullback--Leiblerdivergence of the posterior of each approach with respect to the posteriorassociated to the best model. Our bounds lead to criteria for choosingobservation operators that mitigate the effect of model error for Bayesianinverse problems of this type. We illustrate the feasibility of one suchcriterion on an advection-diffusion-reaction PDE inverse problem, and use thisexample to discuss the importance and challenges of model error-awareinference.

Choosing observation operators to mitigate model error in Bayesian inverse problems