Particle Filters — A Theoretical Perspective

The purpose of this chapter is to present a rigorous mathematical treatment of the convergence of particle filters. In general, we follow the notation and settings suggested by the editors, any extra notation being defined in the next section. Section 2.3.1 contains the main results of the paper: Theorems 2.3.1 and 2.3.2 provide necessary and sufficient conditions for the convergence of the particle filter to the posterior distribution of the signal. As an application of these results, we prove the convergence of a certain class of particle filters. This class includes several known filters (such as those presented in (Carpenter, Clifford and Fearnhead 1999b, Crisan, Del Moral and Lyons 1999, Gordon et al. 1993), but is by no means the most general one. Finally, we discuss some of the issues that are relevant in applications and which arise from the theoretical analysis of these methods.