  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.
