Sequential Monte Carlo state estimation for Poisson observations at discrete times

We consider estimating a stochastic intensity process, indirectly observed through a sequence of Poisson random variables. The class of dynamical systems we consider, are autoregressive state processes and a sequence of Poisson distributed observations, each influenced by the state process. Computing explicit recursive filters for this class of system can be technically difficult. Further, exact filters for this class of model do not have fixed memory requirements. It is shown in Manton et al. (1999), that the exact filter for these models has linear growth with respect to time in the number of sufficient statistics needed to construct the filter density. Some approximate suboptimal filters are available for these models using Edgeworth expansions. Using Sequential Monte Carlo (SMC) methods and the so called Bayesian bootstrap filter, we propose a scheme to estimate an intensity process for the models just described. The scheme we present is simple to implement and has fixed memory requirements. A simulation study is included to demonstrate the performance of SMC methods for discrete time models with Poisson observations.