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.

翻译：在统计推断中，生成数据的参数到可观测映射与用于推断的参数到可观测映射之间的差异可能导致似然函数误设，从而产生不正确的估计。在许多反问题中，参数到可观测映射是线性状态到可观测映射（称为“观测算子”）与可能非线性的参数到状态映射（称为“模型”）的复合。我们考虑此类贝叶斯反问题，其中参数到可观测映射的差异源于使用了与最优模型不同的近似模型，即非零的“模型误差”。已有多种方法被提出以处理此类差异，每种方法都会产生一个特定的后验分布。我们展示了如何利用后验分布关于似然扰动的局部Lipschitz稳定性估计，来界定每种方法的后验分布相对于最优模型对应后验分布的Kullback-Leibler散度。我们的界导出了观测算子的选择准则，能够减轻此类贝叶斯反问题中模型误差的影响。我们通过对一个对流-扩散-反应偏微分方程反问题应用其中一个准则，验证了其可行性，并以此例讨论了模型误差感知推断的重要性和挑战。