Estimating the statistics of the state of a dynamical system, from partial and noisy observations, is both mathematically challenging and finds wide application. Furthermore, the applications are of great societal importance, including problems such as probabilistic weather forecasting and prediction of epidemics. Particle filters provide a well-founded approach to the problem, leading to provably accurate approximations of the statistics. However these methods perform poorly in high dimensions. In 1994 the idea of ensemble Kalman filtering was introduced by Evensen, leading to a methodology that has been widely adopted in the geophysical sciences and also finds application to quite general inverse problems. However, ensemble Kalman filters have defied rigorous analysis of their statistical accuracy, except in the linear Gaussian setting. In this article we describe recent work which takes first steps to analyze the statistical accuracy of ensemble Kalman filters beyond the linear Gaussian setting. The subject is inherently technical, as it involves the evolution of probability measures according to a nonlinear and nonautonomous dynamical system; and the approximation of this evolution. It can nonetheless be presented in a fairly accessible fashion, understandable with basic knowledge of dynamical systems, numerical analysis and probability.

翻译：从部分含噪观测中估计动态系统状态的统计量，既面临数学挑战又具有广泛的应用价值。此类应用涉及重大社会意义，包括概率天气预报和流行病预测等问题。粒子滤波为该问题提供了严谨的方法论基础，能够生成可证明精度的状态统计量近似值，但在高维场景下表现欠佳。1994年Evensen提出的集合卡尔曼滤波方法开创了新的技术路径，该技术在地球物理科学领域被广泛应用，并拓展至诸多通用反演问题。然而除线性高斯场景外，集合卡尔曼滤波的统计精度一直缺乏严格的理论分析。本文综述了近期突破线性高斯假设限制、率先开展集合卡尔曼滤波精度分析的研究成果。该课题本质具有高度技术性，涉及根据非线性非自治动力系统演化概率测度及其近似逼近过程，但通过动态系统、数值分析和概率论的基础知识即可较为直观地理解相关结论。