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：贝叶斯逆问题的建模涉及先验分布的选择，看似同样合理的不同选择可能导致显著不同的结论。我们发展了一种计算方法，以更好地理解定义先验的超参数对感兴趣量的后验统计量的影响。该方法依赖于贝叶斯逆问题对定义先验的超参数的全局敏感性分析（GSA）。然而，这是一个具有挑战性的问题——朴素的二重循环采样方法需要运行大量马尔可夫链蒙特卡洛（MCMC）采样程序。本工作通过（i）高效代理模型的巧妙结合与（ii）定制化重要性采样方法，为这种敏感性分析的实践应用奠定了关键基础。特别地，我们无需重复执行MCMC运行，即可准确开展后验预测统计量对先验超参数的全局敏感性分析。我们在一个简单贝叶斯线性逆问题及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。