Based on Bellman's dynamic-programming principle, Lange (2024) presents an approximate method for filtering, smoothing and parameter estimation for possibly non-linear and/or non-Gaussian state-space models. While the approach applies more generally, this pedagogical note highlights the main results in the case where (i) the state transition remains linear and Gaussian while (ii) the observation density is log-concave and sufficiently smooth in the state variable. I demonstrate how Kalman's (1960) filter and Rauch et al.'s (1965) smoother can be obtained as special cases within the proposed framework. The main aim is to present non-experts (and my own students) with an accessible introduction, enabling them to implement the proposed methods.

翻译：基于贝尔曼动态规划原理，Lange（2024）提出了一种适用于可能非线性、非高斯状态空间模型的滤波、平滑与参数估计的近似方法。尽管该方法具有更广泛的适用性，本教学注释重点阐述了以下两种情形下的主要结论：（i）状态转移保持线性高斯特性；（ii）观测密度关于状态变量呈对数凹且充分光滑。我论证了卡尔曼（1960）滤波器与劳赫等人（1965）平滑器可视为该框架下的特例。本文主要目标是为非专业人士（及本人学生）提供一份易于理解的入门介绍，使其能够实现所提方法。