  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.
