We study the celebrated Shiryaev-Wonham filter in its historical setup of Wonham (1964) where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of the authors show the appearance of spikes in the filtered process, akin to a metastability phenomenon. This paper is aimed at understanding the smoothed optimal filter, which is relevant for any system with feedback. In particular, we demonstrate that there is a sharp phase transition between a spiking regime and a regime with perfect smoothing.

翻译：我们研究经典Shiryaev-Wonham滤波器在其历史框架Wonham(1964)中的表现，其中隐藏马尔可夫跳变过程具有两种状态。我们关注观测方程的弱噪声区域。有趣的是，这对应滤波方程的强噪声区域。作者先前的研究结果表明，滤波过程中会出现类似于亚稳态现象的尖峰。本文旨在理解与任何反馈系统相关的平滑最优滤波器。特别地，我们证明在尖峰区域与完全平滑区域之间存在尖锐的相变。