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

Abstract

Particle filters (PFs) form a class of Monte Carlo algorithms that propagate over time a set of $N\ge 1$ particles which can be used to estimate, in an online fashion, the sequence of filtering distributions $({\hat{\eta }}_t{)}_{t\ge 1}$ 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 $({\hat{\eta }}_t^N{)}_{t\ge 1}$ the PF estimate of $({\hat{\eta }}_t{)}_{t\ge 1}$ and letting $\kappa \in (0,1/2)$, in this work we first show that for any number of particles $N$ it holds that, with probability one, we have $\Vert {\hat{\eta }}_t^N-{\hat{\eta }}_t\Vert \ge \kappa$ for infinitely many time instants $t\ge 1$, with $\Vert \cdot \Vert$ 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 $\{{\hat{\eta }}_t{\}}_{t=1}^T$ of filtering distributions by studying the probability $\mathbb{P}({\sup }_{t\in \{1,\dots ,T\}}\Vert {\hat{\eta }}_t^N-{\hat{\eta }}_t\Vert \ge \kappa )$. 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 ${\lim }_{N\to \infty }{\sup }_{t\ge 1}\Vert {\hat{\eta }}_t^N-{\hat{\eta }}_t\Vert =0$ with probability one.

24 pages (major revision of the paper)