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

Abstract: Particle filters (PFs) is a class of Monte Carlo algorithms that propagate over time a set of $N\in\mathbb{N}$ particles which can be used to estimate, in an online fashion, the sequence of filtering distributions $(\hat{\eta}t){t\geq 1}$ defined by a state-space model. Despite the popularity of PFs, the study of the time evolution of their estimates has received barely attention in the literature. Denoting by $(\hat{\eta}t^N){t\geq 1}$ the PF estimate of $(\hat{\eta}t){t\geq 1}$ and letting $\kappa\in (0,1)$, in this work we first show that for any number of particles $N$ it holds that, with probability one, we have $|\hat{\eta}t^N- \hat{\eta}_t|\geq \kappa$ for infinitely many $t\geq 1$, with $|\cdot|$ a measure of 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 $\mathbb{P}(\sup_{t\in{1,\dots,T}}|\hat{\eta}t^{N}-\hat{\eta}_t|\geq \kappa)$. 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 $\lim{N\rightarrow\infty}\sup_{t\geq 1}|\hat{\eta}_t^N-\hat{\eta}_t|=0$ with probability one.