  We consider the task of filtering a dynamic parameter evolving as a diffusion
process, given data collected at discrete times from a likelihood which is
conjugate to the marginal law of the diffusion, when a generic dual process on
a discrete state space is available. Recently, it was shown that duality with
respect to a death-like process implies that the filtering distributions are
finite mixtures, making exact filtering and smoothing feasible through
recursive algorithms with polynomial complexity in the number of observations.
Here we provide general results for the case of duality between the diffusion
and a regular jump continuous-time Markov chain on a discrete state space,
which typically leads to filtering distribution given by countable mixtures
indexed by the dual process state space. We investigate the performance of
several approximation strategies on two hidden Markov models driven by
Cox-Ingersoll-Ross and Wright-Fisher diffusions, which admit duals of
birth-and-death type, and compare them with the available exact strategies
based on death-type duals and with bootstrap particle filtering on the
diffusion state space as a general benchmark.
