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 the feasibility of one such criterion on an advection-diffusion-reaction PDE inverse problem, and use this example to discuss the importance and challenges of model error-aware inference.

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