The Unscented Rao-Blackwellized Marginal Particle Filter applied to the joint state and parameter estimation problem

1 Problem mathematical formulation Let us consider a time-varying system described by k, θ, Sk, Zk, where k is the time index, θ is a vector of parameters, Sk is a random vector corresponding to the time-varying state at time k (with the corresponding realization denoted by sk), and Zk is a random vector corresponding to the observation process at time k (with the corresponding realization denoted by zk). We assume that the time evolution of the system is characterized by sk+1 = fk (sk, θ, ξ s k) (1) zk = hk (sk, θ, ξ z k) (2) s0 ∼ p(s0) (3) where fk and hk are arbitrary nonlinear functions, and (Ξsk) ∞ k=0 and (Ξ z k) ∞ k=1 (with realizations denoted by (ξ s k) ∞ k=0 and (ξ k) ∞ k=1) are mutually independent and time-independent noise sequences, also independent from p(s0). Our goal is to obtain estimates θ̂ and ŝk respectively of θ and Sk, given all available observations Z , (z1, . . . , zk). We tackle the problem using a coupled Bayesian strategy, where states and parameters are treated as a single augmented state [ S k ,Θ T T , with realizations given by [ sTk , θ T T , and the statistical information about the augmented state summarized by the joint posterior density p (