Optimal SIR algorithm vs. fully adapted auxiliary particle filter: a non asymptotic analysis

Particle filters (PF) and auxiliary particle filters (APF) are widely used sequential Monte Carlo (SMC) techniques. In this paper we comparatively analyse, from a non asymptotical point of view, the Sampling Importance Resampling (SIR) PF with optimal conditional importance distribution (CID) and the fully adapted APF (FA). We compute the (finite samples) conditional second order moments of Monte Carlo (MC) estimators of a moment of interest of the filtering pdf, and analyse under which circumstances the FA-based estimator outperforms (or not) the optimal Sequential Importance Sampling (SIS)-based one. Our analysis is local, in the sense that we compare the estimators produced by one time step of the different SMC algorithms, starting from a common set of weighted points. This analysis enables us to propose a hybrid SIS/FA algorithm which automatically switches at each time step from one loop to the other. We finally validate our results via computer simulations.