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. 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 demonstrate that if the true filtering distribution is close to Gaussian in the joint space of state and data, in the weighted total variation metric, then the true-filter is well approximated by the ensemble Kalman filter, in the same metric. Finally, we 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，它具有粒子滤波等算法所不具备的鲁棒性，尤其不会出现权重坍塌现象。然而，除高斯场景外，尚无理论能量化其作为真实滤波分布近似解的精度。为解决这一问题，我们首次对高斯场景以外的集合卡尔曼滤波精度进行了分析。我们的分析聚焦于均值场集合卡尔曼滤波。我们将该滤波器的更新方程以及真实滤波分布的更新方程重新表述为概率测度上的映射形式。引入加权全变差度量来估计两种滤波器的距离，并针对该度量下定义两种滤波器演化的映射，证明了多种稳定性估计。利用这些稳定性估计，我们证明：若在状态与数据的联合空间中，真实滤波分布在加权全变差度量下接近高斯分布，则在同一度量下，集合卡尔曼滤波能良好近似该真实滤波分布。最后，我们将这些结果推广至高斯投影滤波器——它可视为无迹卡尔曼滤波的均值场描述。