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We study the celebrated Shiryaev-Wonham filter in its historical setup ofWonham (1964) where the hidden Markov jump process has two states. We areinterested 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 filteredprocess, akin to a metastability phenomenon. This paper is aimed atunderstanding the smoothed optimal filter, which is relevant for any systemwith feedback. In particular, we demonstrate that there is a sharp phasetransition between a spiking regime and a regime with perfect smoothing.

To spike or not to spike: the whims of the Wonham filter in the strong noise regime