Further Rao-Blackwellizing an already Rao-Blackwellized algorithm for Jump Markov State Space Systems

Exact Bayesian filtering is impossible in Jump Markov State Space Systems (JMSS), even in the simple linear and Gaussian case. Suboptimal solutions include sequential Monte-Carlo (SMC) algorithms which are indeed popular, and are declined in different versions according to the JMSS considered. In particular, Jump Markov Linear Systems (JMLS) are particular JMSS for which a Rao-Blackwellized (RB) Particle Filter (PF) has been derived. The RBPF solution relies on a combination of PF and Kalman Filtering (KF), and RBPF-based moment estimators outperform purely SMC-based ones when the number of samples tends to infinity. In this paper, we show that it is possible to derive a new RBPF solution, which implements a further RB step in the already RBPF with optimal importance distribution (ID). The new RBPF-based moment estimator outperforms the classical RBPF one whatever the number of particles, at the expense of a reasonable extra computational cost.