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Central Limit Theorems for Coupled Particle Filters

Ajay Jasra, Fangyuan Yu

TL;DR

This paper develops central limit theorems for coupled particle filters (CPFs) that estimate differences between two close filters and introduces a new CPF (MCPF) that approximates the maximal coupling of predictor distributions. It analyzes four CPFs (IRCPF, MRCPF, MCPF, WCPF) and proves CLTs for the first three, along with general asymptotic-variance bounds; under suitable assumptions, the MCPF and WCPF achieve time-uniform variance bounds in partially observed diffusions with Euler discretization, preserving the forward-rate of the diffusion. The authors show that naive independent resampling (IRCPF) fails to maintain time-uniform accuracy, while MRCPF can exhibit unfavorable exponential-in-time variance growth in general. A detailed PODDP MLMC application establishes how discretization and coupling interact to control variance across levels, with explicit bounds for the MCPF and WCPF that scale with $Δ_l$ (or $(Δ_l)^{1/(1+λ)}$) and depend on Wasserstein-type couplings. An extensive example verifies the assumptions in a one-dimensional diffusion setting, highlighting the practical viability and limitations of the proposed methods for efficient multilevel estimation in diffusion models.

Abstract

In this article we prove a new central limit theorem (CLT) for coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations w.r.t. filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with discretization $Δ_l$ we show that the MCPF and the approach in Jasra et al. (2018) have, under assumptions, an asymptotic variance that is upper-bounded by an expression that is (almost) $\mathcal{O}(Δ_l)$, uniformly in time. The $\mathcal{O}(Δ_l)$ rate preserves the so-called forward rate of the diffusion in some scenarios which is not the case for the CPF in Jasra et al (2017).

Central Limit Theorems for Coupled Particle Filters

TL;DR

This paper develops central limit theorems for coupled particle filters (CPFs) that estimate differences between two close filters and introduces a new CPF (MCPF) that approximates the maximal coupling of predictor distributions. It analyzes four CPFs (IRCPF, MRCPF, MCPF, WCPF) and proves CLTs for the first three, along with general asymptotic-variance bounds; under suitable assumptions, the MCPF and WCPF achieve time-uniform variance bounds in partially observed diffusions with Euler discretization, preserving the forward-rate of the diffusion. The authors show that naive independent resampling (IRCPF) fails to maintain time-uniform accuracy, while MRCPF can exhibit unfavorable exponential-in-time variance growth in general. A detailed PODDP MLMC application establishes how discretization and coupling interact to control variance across levels, with explicit bounds for the MCPF and WCPF that scale with (or ) and depend on Wasserstein-type couplings. An extensive example verifies the assumptions in a one-dimensional diffusion setting, highlighting the practical viability and limitations of the proposed methods for efficient multilevel estimation in diffusion models.

Abstract

In this article we prove a new central limit theorem (CLT) for coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations w.r.t. filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with discretization we show that the MCPF and the approach in Jasra et al. (2018) have, under assumptions, an asymptotic variance that is upper-bounded by an expression that is (almost) , uniformly in time. The rate preserves the so-called forward rate of the diffusion in some scenarios which is not the case for the CPF in Jasra et al (2017).

Paper Structure

This paper contains 33 sections, 45 theorems, 396 equations.

Table of Contents

  1. Introduction
  2. Notation and Models
  3. Notations
  4. Models
  5. Example
  6. Multilevel Monte Carlo
  7. Algorithms
  8. Independent Pair Resampling
  9. Maximally Coupled Resampling
  10. Maximal Coupling
  11. Wasserstein Coupled Resampling
  12. Theoretical Results
  13. Central Limit Theorems
  14. Asymptotic Variance
  15. Application to Partially Observed Diffusions
  16. ...and 18 more sections

Key Result

Theorem 3.1

Assume (Ahyp:1-hyp:3). Then for any $\varphi\in\mathcal{C}_b(\mathsf{X}\times\mathsf{X})$, $n\geq 0$ we have

Theorems & Definitions (94)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.1
  • Theorem 4.3
  • Remark 4.2
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • ...and 84 more