Safety of particle filters: Some results on the time evolution of particle filter estimates

Particle filters (PFs) form a class of Monte Carlo algorithms that propagate over time a set of N≥1 particles which can be used to estimate, in an online fashion, the sequence of filtering distributions (ηˆt)t≥1 defined by a state-space model. Despite the popularity of PFs, the study of the time evolution of their estimates has received barely any attention in the literature. Denoting by (ηˆtN)t≥1 the PF estimate of (ηˆt)t≥1 and letting κ∈(0,1∕2), in this work we first show that for any number of particles N it holds that, with probability one, we have ‖ηˆtN−ηˆt‖≥κ for infinitely many time instants t≥1, with ‖⋅‖ the Kolmogorov distance between probability distributions. Considering a simple filtering problem we then provide reassuring results concerning the ability of PFs to estimate jointly a finite set {ηˆt}t=1T of filtering distributions by studying the probability P(supt∈{1,…,T}‖ηˆtN−ηˆt‖≥κ). Finally, on the same toy filtering problem, we prove that sequential quasi-Monte Carlo, a randomized quasi-Monte Carlo version of PF algorithms, offers greater safety guarantees than PFs in the sense that, for this algorithm, it holds that limN→∞supt≥1‖ηˆtN−ηˆt‖=0 with probability one.