Safety of particle filters: Some results on the time evolution of particle filter estimates
Mathieu Gerber
TL;DR
The paper analyzes the time evolution of particle filter estimates in a one-dimensional linear Gaussian state-space model with zero observations, revealing that PFs cannot maintain time-uniform accuracy: for any $N$, with probability one there are infinitely many times where the Kolmogorov distance between the PF estimate $\hat{\eta}_t^N$ and the true $\hat{\eta}_t$ exceeds a fixed $\kappa\in(0,1/2)$. It then derives a sharp finite-horizon bound showing that the probability of large error over $t\in\{1,\dots,T\}$ can be controlled by choosing $N$ growing like $N\ge C v_\kappa \log(T/q)$ with $v_\kappa=(1+\log(1+\kappa^{-1}))^2 \kappa^{-2}$. To address safety concerns, the authors introduce sequential quasi-Monte Carlo (SQMC) and prove that, for the same toy model, $\lim_{N\to\infty} \sup_{t\ge1} \|\tilde{\eta}_t^N-\hat{\eta}_t\|=0$ almost surely, providing time-uniform, almost-sure convergence guarantees. The results are established through a general discrepancy bound for filtering, DKW-based concentration, and novel KH/mixture-discrepancy arguments, complemented by a detailed stability analysis of the underlying model. Overall, the work quantifies the time-evolution safety of PFs and demonstrates how de-randomization via SQMC can restore time-uniform convergence in a principled manner, with explicit bounds and sharp dependence on horizon and confidence level.
Abstract
Particle filters (PFs) form a class of Monte Carlo algorithms that propagate over time a set of $N\geq 1$ particles which can be used to estimate, in an online fashion, the sequence of filtering distributions $(\hatη_t)_{t\geq 1}$ defined by a state-space model. Despite the popularity of PFs, the study of the time evolution of their estimates has received barely any attention in the literature. Denoting by $(\hatη_t^N)_{t\geq 1}$ the PF estimate of $(\hatη_t)_{t\geq 1}$ and letting $κ\in (0,1/2)$, in this work we first show that for any number of particles $N$ it holds that, with probability one, we have $\|\hatη_t^N- \hatη_t\|\geq κ$ for infinitely many time instants $t\geq 1$, with $\|\cdot\|$ 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 $\{\hatη_t\}_{t=1}^T$ of filtering distributions by studying the probability $\mathbb{P}(\sup_{t\in\{1,\dots,T\}}\|\hatη_t^{N}-\hatη_t\|\geq κ)$. 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 $\lim_{N\rightarrow\infty}\sup_{t\geq 1}\|\hatη_t^N-\hatη_t\|=0$ with probability one.