Many parameter estimation problems arising in applications can be cast in the framework of Bayesian inversion. This allows not only for an estimate of the parameters, but also for the quantification of uncertainties in the estimates. Often in such problems the parameter-to-data map is very expensive to evaluate, and computing derivatives of the map, or derivative-adjoints, may not be feasible. Additionally, in many applications only noisy evaluations of the map may be available. We propose an approach to Bayesian inversion in such settings that builds on the derivative-free optimization capabilities of ensemble Kalman inversion methods. The overarching approach is to first use ensemble Kalman sampling (EKS) to calibrate the unknown parameters to fit the data; second, to use the output of the EKS to emulate the parameter-to-data map; third, to sample from an approximate Bayesian posterior distribution in which the parameter-to-data map is replaced by its emulator. This results in a principled approach to approximate Bayesian inference that requires only a small number of evaluations of the (possibly noisy approximation of the) parameter-to-data map. It does not require derivatives of this map, but instead leverages the documented power of ensemble Kalman methods. Furthermore, the EKS has the desirable property that it evolves the parameter ensemble towards the regions in which the bulk of the parameter posterior mass is located, thereby locating them well for the emulation phase of the methodology. In essence, the EKS methodology provides a cheap solution to the design problem of where to place points in parameter space to efficiently train an emulator of the parameter-to-data map for the purposes of Bayesian inversion.

在应用程序中出现的许多参数估计问题都可以在贝叶斯反演的框架中进行。这不仅可以估计参数，还可以量化估计中的不确定性。通常，在此类问题中，参数到数据映射的评估非常昂贵，并且计算映射的导数或导数伴随可能不可行。另外，在许多应用中，可能只有地图的嘈杂评估。我们提出了一种在此类设置中基于集合卡尔曼反演方法的无导数优化功能的贝叶斯反演方法。总体方法是首先使用集合卡尔曼采样（EKS）来校准未知参数以适合数据；第二，使用EKS的输出来模拟参数到数据映射；第三，取样从近似贝叶斯后验分布中，其中参数到数据的映射被其仿真器替代。这导致了一种近似贝叶斯推断的原则方法，该方法仅需要对参数到数据映射（可能是噪声近似）进行少量评估。它不需要此地图的派生词，而是利用了集成Kalman方法的已记录功能。此外，EKS具有理想的特性，它将参数集合朝着大部分参数后质量所在的区域发展，从而在方法的仿真阶段很好地定位它们。在本质上，