The ensemble Kalman filter is widely used in applications because, for high dimensional filtering problems, it has a robustness that is not shared for example by the particle filter; in particular it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. We prove two types of results: the first type comprise a stability estimate controlling the error made by the ensemble Kalman filter in terms of the difference between the true filtering distribution and a nearby Gaussian; and the second type use this stability result to show that, in a neighbourhood of Gaussian problems, the ensemble Kalman filter makes a small error, in comparison with the true filtering distribution. Our analysis is developed for the mean field ensemble Kalman filter. We rewrite the update equations for this filter, and for the true filtering distribution, in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters and we prove various stability estimates for the maps defining the evolution of the two filters, in this metric. Using these stability estimates we prove results of the first and second types, in the weighted total variation metric. We also provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean field description of the unscented Kalman filter.

翻译：集合卡尔曼滤波在应用中广泛使用，因为对于高维滤波问题，它具有粒子滤波等不具备的稳健性，尤其不会出现权重坍塌问题。然而，除高斯设定外，尚无理论量化其作为真实滤波分布近似的精度。为解决这一问题，我们首次对高斯设定之外的集合卡尔曼滤波精度进行了分析。我们证明了两类结果：第一类包括稳定性估计，通过真实滤波分布与邻近高斯分布之间的差异控制集合卡尔曼滤波的误差；第二类利用这一稳定性估计表明，在高斯问题邻域内，集合卡尔曼滤波相较于真实滤波分布产生的误差较小。我们的分析针对平均场集合卡尔曼滤波展开。我们将该滤波器和真实滤波分布的更新方程重写为概率测度上的映射形式。引入加权全变差度量来估计两个滤波器之间的距离，并在此度量下证明定义两个滤波器演化的映射的各种稳定性估计。利用这些稳定性估计，我们在加权全变差度量下证明了第一类和第二类结果。我们还将这些结果推广到高斯投影滤波（可视为无迹卡尔曼滤波的平均场描述）。