A general smoothing equation for Poisson observations

We compute a general smoothing equation for a doubly stochastic Poisson process (DSPP) whose intensity is influenced by a discrete state Markov process. This equation can be readily applied to specific smoothing algorithms referred to in the signal processing literature as fixed point smoothing, fixed lag smoothing and fixed interval smoothing. To compute our smoothing equation, we begin with the observation-parametrised dynamics for a gauge transformed unnomalised probability vector. By appealing to a duality between forwards and backwards processes, we derive an observation-parametrised equation for a backwards dynamical system. Smoothed posterior probabilities are then obtained by combining the solutions of the forward and backwards observation-parametrised equations. A computer simulation is included to demonstrate the benefits of smoothing over filtering in a fixed interval smoothing scenario