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On the Forgetting of Particle Filters

Joona Karjalainen, Anthony Lee, Sumeetpal S. Singh, Matti Vihola

TL;DR

This work shows that if the underlying Feynman–Kac model satisfies a strong mixing condition, the particle-filter state, viewed as an inhomogeneous Markov chain, forgets its initial condition at an exponential rate in total variation, requiring only $O(\log N)$ algorithm steps for forgetting. The authors extend the forgetting analysis to the conditional particle filter (CPF), obtaining analogous $O(\log N)$ forgetting along with time-uniform $L^p$ bounds and new propagation-of-chaos results. They furthermore develop coupling mechanisms and discuss implications for algorithms that couple particle filters, including applications to out-of-sequence measurements. The results substantially improve on prior conservative bounds, provide sharp rates, and offer practical insights for coupling, smoothing, and OOS scenarios in sequential Monte Carlo methods.

Abstract

We study the forgetting properties of the particle filter when its state - the collection of particles - is regarded as a Markov chain. Under a strong mixing assumption on the particle filter's underlying Feynman-Kac model, we find that the particle filter is exponentially mixing, and forgets its initial state in $O(\log N )$ 'time', where $N$ is the number of particles and time refers to the number of particle filter algorithm steps, each comprising a selection (or resampling) and mutation (or prediction) operation. We present an example which shows that this rate is optimal. In contrast to our result, available results to-date are extremely conservative, suggesting $O(α^N)$ time steps are needed, for some $α>1$, for the particle filter to forget its initialisation. We also study the conditional particle filter (CPF) and extend our forgetting result to this context. We establish a similar conclusion, namely, CPF is exponentially mixing and forgets its initial state in $O(\log N )$ time. To support this analysis, we establish new time-uniform $L^p$ error estimates for CPF, which can be of independent interest. We also establish new propagation of chaos type results using our proof techniques, discuss implications to couplings of particle filters and an application to processing out-of-sequence measurements.

On the Forgetting of Particle Filters

TL;DR

This work shows that if the underlying Feynman–Kac model satisfies a strong mixing condition, the particle-filter state, viewed as an inhomogeneous Markov chain, forgets its initial condition at an exponential rate in total variation, requiring only algorithm steps for forgetting. The authors extend the forgetting analysis to the conditional particle filter (CPF), obtaining analogous forgetting along with time-uniform bounds and new propagation-of-chaos results. They furthermore develop coupling mechanisms and discuss implications for algorithms that couple particle filters, including applications to out-of-sequence measurements. The results substantially improve on prior conservative bounds, provide sharp rates, and offer practical insights for coupling, smoothing, and OOS scenarios in sequential Monte Carlo methods.

Abstract

We study the forgetting properties of the particle filter when its state - the collection of particles - is regarded as a Markov chain. Under a strong mixing assumption on the particle filter's underlying Feynman-Kac model, we find that the particle filter is exponentially mixing, and forgets its initial state in 'time', where is the number of particles and time refers to the number of particle filter algorithm steps, each comprising a selection (or resampling) and mutation (or prediction) operation. We present an example which shows that this rate is optimal. In contrast to our result, available results to-date are extremely conservative, suggesting time steps are needed, for some , for the particle filter to forget its initialisation. We also study the conditional particle filter (CPF) and extend our forgetting result to this context. We establish a similar conclusion, namely, CPF is exponentially mixing and forgets its initial state in time. To support this analysis, we establish new time-uniform error estimates for CPF, which can be of independent interest. We also establish new propagation of chaos type results using our proof techniques, discuss implications to couplings of particle filters and an application to processing out-of-sequence measurements.
Paper Structure (14 sections, 33 theorems, 178 equations, 1 figure, 3 algorithms)

This paper contains 14 sections, 33 theorems, 178 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and particle filter definition
  3. Forgetting of the particle filter
  4. Relationship with existing forgetting result
  5. Proof of Theorem \ref{['thm:pfforget']}
  6. An example where $\log(N)$ steps is necessary and sufficient
  7. Forgetting of the conditional particle filter
  8. $L^p$ bounds for the conditional particle filter
  9. Connections to propagation of chaos
  10. Implications for algorithms that couple particle filters
  11. Application to out-of-sequence measurements
  12. Discussion
  13. Technical lemmas
  14. On weakening the assumptions (A1)

Key Result

Lemma 1

(del2004feynman, Proposition 4.3.6) Assume (A1). For all probability measures $\mu$ and $\nu$, $n\geq 0$ and $k\geq 0$, where $\beta = 1-(\underline M / \bar{M})^2$.

Figures (1)

  • Figure 1: Propagation of chaos total variation distances \ref{['eq:poc-bound-hellinger']} as a function of $q/N$ in a discrete state-space model example for times $k=4$ and $k=20$. The lines illustrate the mentioned orders of decay.

Theorems & Definitions (73)

  • Definition
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Definition
  • ...and 63 more