[Paper Review] Smoothing with Couplings of Conditional Particle Filters

Pierre Jacob, Fredrik Lindsten+1 more|arXiv (Cornell University)|Jan 8, 2017

Advanced Statistical Process Monitoring4 citations

TL;DR

This paper proposes an unbiased smoothing estimator for state space models by combining conditional particle filters with Rhee–Glynn debiasing techniques. The method enables parallel computation and valid confidence intervals via independent estimators, achieving manageable variance increase despite bias removal, as validated in experiments on toy models and a Lotka-Volterra system with intractable transitions.

ABSTRACT

In state space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits: independent estimators can be generated in parallel, and confidence intervals can be constructed from the central limit theorem to quantify the approximation error. To design unbiased estimators, we combine a generic debiasing technique for Markov chains with a Markov chain Monte Carlo algorithm for smoothing. The resulting procedure is widely applicable and we show in numerical experiments that the removal of the bias comes at a manageable increase in variance. We establish the validity of the proposed estimators under mild assumptions. Numerical experiments are provided on toy models, including a setting of highly-informative observations, and a realistic Lotka-Volterra model with an intractable transition density.

Motivation & Objective

• To develop an unbiased estimator for smoothing expectations in general state space models, overcoming the bias inherent in standard particle smoothers.
• To enable parallel generation of independent estimators for improved computational efficiency and statistical inference.
• To provide valid confidence intervals for smoothing estimates using the central limit theorem, which is not possible with biased estimators.
• To maintain computational feasibility and variance control through tuning parameters and averaging over independent replicates.
• To extend the applicability of debiasing techniques to smoothing problems in nonlinear, non-Gaussian state space models.

Proposed method

• The method combines conditional particle filters (CPFs) with the Rhee–Glynn debiasing technique for Markov chains to produce unbiased estimators of smoothing expectations.
• It uses coupled CPF chains that couple trajectories at a random meeting time, ensuring unbiasedness through the Rhee–Glynn framework.
• The estimator is constructed by generating multiple independent pairs of coupled CPF chains and applying the Rhee–Glynn sum to eliminate bias.
• Meeting times between coupled chains are used to determine stopping times, with the coupling time distribution influencing variance and computational cost.
• The approach is validated under mild conditions, including finiteness of computational cost and variance, ensuring theoretical validity.
• Tuning parameters such as particle count $N$, coupling lag $k$, and meeting time threshold $m$ are used to control variance and computational load.

Experimental results

Research questions

• RQ1 Can unbiased smoothing estimators be constructed for general state space models using particle filtering and debiasing techniques?
• RQ2 Does the proposed method enable reliable confidence intervals and parallel computation, unlike standard biased smoothers?
• RQ3 How does the variance of the proposed unbiased estimator compare to that of standard particle filters and fixed-lag smoothers under comparable computational cost?
• RQ4 Can the method be applied to models with intractable transition densities, such as the Lotka-Volterra system?
• RQ5 What is the impact of coupling parameters (e.g., $k$, $m$) on the variance and computational efficiency of the estimator?

Key findings

• The proposed unbiased estimator achieves a relative variance of smoothing mean estimates that is comparable to particle filters with significantly more particles—equivalent to 78,377 particles for $N=4,096$ in the experiment.
• The method reduces variance compared to standard particle filters when using equivalent computational cost, particularly when combined with fixed-lag smoothing.
• Meeting time distributions from 1,000 independent chains with $N=4,096$ particles were used to set $k=7$, $m=14$, indicating effective coupling.
• The bias of fixed-lag smoothing was found to be negligible in the zooplankton model, but such bias is hard to assess with fixed-lag methods alone.
• The method maintains theoretical validity under mild conditions, including finite computational cost and finite variance of the estimator.
• The framework is extensible to joint parameter and state inference via particle MCMC and could support perfect simulation or score estimation via Fisher’s identity.