We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by PDEs with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on a model inverse problem governed by a PDE for heat conduction.

翻译：我们考虑了非线性贝叶斯人非线性敏感度分析(HDSA)对由无限参数参数的PDE所管辖的非线性贝叶斯人的问题的高度差异性敏感度分析(HDSA),在以往的著作中,HDSA被用来评估确定性反问题的解决方案对于其他模型不确定性和不同类型测量数据的敏感性,在目前的工作中,我们将HDSA扩大到PDE所管理的巴伊斯人反问题类别。重点是评估从后方分布中得出的某些关键数量的敏感性。具体地说,我们侧重于分析MAP点和贝斯风险的敏感性,并充分利用Bayesian反问题所包含的信息。在为Bayesian反问题HDSA建立了数学框架之后,我们提出了计算拟议的HDSA指数的详细计算方法。我们研究了拟议对由PDE所管辖的热导模型采取的办法的有效性。