Nested particle filters for online parameter estimation in discrete-time state-space Markov models

We address the problem of estimating the fixed parameters of a state-space dynamic system using sequential Monte Carlo methods. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability law of the static parameters and the dynamic variables of the system of interest, in a vein similar to the recent "sequential Monte Carlo square" (SMC2) algorithm. However, unlike the SMC2 method, the proposed algorithm operates in a purely sequential and recursive manner. In particular, the computational complexity of the recursive steps of the proposed method is constant over time and the scheme for the rejuvenation of the particles in the parameter space is simpler. We prove that the approximations of integrals of real functions with respect to the posterior distribution of the parameters computed using the proposed scheme converge asymptotically in $L_p$, and provide explicit convergence rates in terms of the number of particles. The analysis can be easily extended to obtain the same kind of result for the joint posterior distribution of the parameters and the dynamic variables. We also discuss the effect of the rejuvenation step on the variance of the particle approximation of the posterior mean of the static parameters and propose a simple scheme that leaves the expected value of the sample variance unchanged while increasing the sample diversity (namely, the expected number of distinct particles).