  We link optimal filtering for hidden Markov models to the notion of duality
for Markov processes. We show that when the signal is dual to a process that
has two components, one deterministic and one a pure death process, and with
respect to functions that define changes of measure conjugate to the emission
density, the filtering distributions evolve in the family of finite mixtures of
such measures and the filter can be computed at a cost that is polynomial in
the number of observations. Special cases of our framework include the Kalman
filter, and computable filters for the Cox-Ingersoll-Ross process and the
one-dimensional Wright-Fisher process, which have been investigated before. The
dual we obtain for the Cox-Ingersoll-Ross process appears to be new in the
literature.
