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：从含噪的部分观测中估计动力系统状态的统计量，既在数学上具有挑战性，又具有广泛的应用价值。此外，这些应用具有重大的社会意义，包括概率天气预报和流行病预测等问题。粒子滤波为解决该问题提供了完善的方法，可得到可证明精确的状态统计量近似。然而，这些方法在高维空间中表现不佳。1994年，Evensen提出了集合卡尔曼滤波的概念，该方法已被广泛应用于地球物理科学领域，并可应用于相当一般的逆问题。然而，除线性高斯情景外，集合卡尔曼滤波的统计精度一直缺乏严格分析。本文描述了近期在超越线性高斯情景下分析集合卡尔曼滤波统计精度的初步工作。该主题本质上具有技术性，因为它涉及概率测度根据非线性非自治动力系统的演化，以及该演化的近似。尽管如此，本文仍能以较为易懂的方式呈现，只需具备动力系统、数值分析和概率论的基础知识即可理解。