A non asymptotical analysis of the optimal SIR algorithm vs. the fully adapted auxiliary particle filter

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques for estimating the a posteriori filtering probability density function (pdf) in a Hidden Markov Chain (HMC). These algorithms have been theoretically analysed from an asymptotical statistics perspective. In this paper we provide a non asymptotical, finite number of samples comparative analysis of two particular SMC algorithms : the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID), and the fully adapted APF (FA). Starting from a common set of N particles, we compute closed form expressions of the mean and variance of the empirical Monte Carlo (MC) estimators of a moment of the a posteriori filtering pdf. Both algorithms have the same mean, but in the case where resampling is used, the variance of the SIR algorithm always exceeds that of the FA algorithm.