A semi-exact sequential Monte Carlo filtering algorithm in Hidden Markov Chains

Bayesian filtering is an important issue in Hidden Markov Chains (HMC) models. In many problems it is of interest to compute both the a posteriori filtering pdf at each time instant n and a moment Θ_n thereof. Sequential Monte Carlo (SMC) techniques, which include Particle filtering (PF) and Auxiliary PF (APF) algorithms, propagate a set of weighted particles which approximate that filtering pdf at time n, and then compute a Monte Carlo (MC) estimate of Θ_n. In this paper we show that in models where the so-called Fully Adapted APF (FA-APF) algorithm can be used such as semi-linear Gaussian state-space models, one can compute an estimate of the moment of interest at time n based only on the new observation y_n and on the set of particles at time n - 1. This estimate does not suffer from the extra MC variation due to the sampling of new particles at time n, and is thus preferable to that based on that new set of particles, due to the Rao-Blackwell (RB) theorem. We finally extend our solution to models where the FA-APF cannot be used any longer.