In recent years, Bayesian inference in large-scale inverse problems found in science, engineering and machine learning has gained significant attention. This paper examines the robustness of the Bayesian approach by analyzing the stability of posterior measures in relation to perturbations in the likelihood potential and the prior measure. We present new stability results using a family of integral probability metrics (divergences) akin to dual problems that arise in optimal transport. Our results stand out from previous works in three directions: (1) We construct new families of integral probability metrics that are adapted to the problem at hand; (2) These new metrics allow us to study both likelihood and prior perturbations in a convenient way; and (3) our analysis accommodates likelihood potentials that are only locally Lipschitz, making them applicable to a wide range of nonlinear inverse problems. Our theoretical findings are further reinforced through specific and novel examples where the approximation rates of posterior measures are obtained for different types of perturbations and provide a path towards the convergence analysis of recently adapted machine learning techniques for Bayesian inverse problems such as data-driven priors and neural network surrogates.

翻译：近年来，贝叶斯推断在科学、工程和机器学习领域的大规模反问题中受到了广泛关注。本文通过分析后验测度在似然势和先验测度扰动下的稳定性，探讨了贝叶斯方法的鲁棒性。我们利用一族与最优传输中对偶问题类似的积分概率度量（散度），提出了新的稳定性结果。与前人工作相比，我们的成果在以下三个方面具有独特性：（1）我们构造了适应所研究问题的新积分概率度量族；（2）这些新度量使我们能够以便捷的方式同时研究似然扰动和先验扰动；（3）我们的分析仅要求似然势为局部Lipschitz连续，因此适用于各类非线性反问题。通过针对不同类型扰动获得后验测度逼近率的创新性具体实例，我们的理论发现得到了进一步验证，并为近期应用于贝叶斯反问题的机器学习技术（如数据驱动先验和神经网络替代模型）的收敛性分析提供了路径。