The formulation of Bayesian inverse problems involves choosing prior distributions; choices that seem equally reasonable may lead to significantly different conclusions. We develop a computational approach to better understand the impact of the hyperparameters defining the prior on the posterior statistics of the quantities of interest. Our approach relies on global sensitivity analysis (GSA) of Bayesian inverse problems with respect to the hyperparameters defining the prior. This, however, is a challenging problem--a naive double loop sampling approach would require running a prohibitive number of Markov chain Monte Carlo (MCMC) sampling procedures. The present work takes a foundational step in making such a sensitivity analysis practical through (i) a judicious combination of efficient surrogate models and (ii) a tailored importance sampling method. In particular, we can perform accurate GSA of posterior prediction statistics with respect to prior hyperparameters without having to repeat MCMC runs. We demonstrate the effectiveness of the approach on a simple Bayesian linear inverse problem and a nonlinear inverse problem governed by an epidemiological model

翻译：贝叶斯逆问题的公式化涉及选择先验分布；看似同样合理的不同选择可能导致显著不同的结论。我们开发了一种计算方法，以更好地理解定义先验的超参数对感兴趣量后验统计量的影响。我们的方法依赖于对定义先验的超参数进行贝叶斯逆问题的全局敏感性分析（GSA）。然而，这是一个具有挑战性的问题——一种朴素的嵌套双循环采样方法需要运行数量惊人的马尔可夫链蒙特卡洛（MCMC）采样过程。本工作通过（i）高效替代模型的巧妙组合和（ii）量身定制的重要性采样方法，为实用化此类敏感性分析奠定了基础性步骤。具体而言，我们无需重复运行MCMC即可对先验超参数相关的后验预测统计量进行精确的GSA。我们在一个简单的贝叶斯线性逆问题和一个由流行病学模型控制的非线性逆问题上验证了该方法的有效性。