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.

翻译：在统计推断中，生成数据的参数-观测映射与推断所用的参数-观测映射之间的差异可能导致似然函数设定错误，从而产生不正确的估计。在许多逆问题中，参数-观测映射由两部分组成：一个称为"观测算子"的线性状态-观测映射，以及一个可能是非线性的参数-状态映射（称为"模型"）。我们考虑这类贝叶斯逆问题，其中参数-观测映射的差异源于使用了与最佳模型不同的近似模型，即存在非零的"模型误差"。目前已提出多种方法处理此类差异，每种方法都对应特定的后验分布。我们展示了如何利用后验分布关于似然扰动的局部利普希茨稳定性估计，来界定每种方法的后验分布与最佳模型对应的后验分布之间的库尔贝克-莱布勒散度。这些界限导出了选择观测算子的准则，从而减轻此类贝叶斯逆问题中模型误差的影响。我们以文献中的一个对流-扩散-反应偏微分方程逆问题为例，阐释了其中一种准则，并借此讨论模型误差感知推断的重要性与挑战。