Smoothing With Couplings of Conditional Particle Filters

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 CI 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 for a realistic Lotka–Volterra model with an intractable transition density. Supplementary materials for this article are available online.