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 the first unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits, as it allows for a complete parallelization of the algorithm and for computing accurate confidence intervals. The method combines two recent breakthroughs: the first is a generic debiasing technique for Markov chains due to Rhee and Glynn, and the second is the introduction of a uniformly ergodic Markov chain for smoothing, the conditional particle filter of Andrieu, Doucet and Holenstein. We show how a combination of the two methods delivers practical estimators, upon the introduction of couplings between conditional particle filters. The algorithm is widely applicable, has minimal tuning parameters and is amenable to modern computing hardware. We establish the validity of the proposed estimator under mild assumptions. Numerical experiments illustrate its performance in a toy model and in a Lotka-Volterra model with an intractable transition density.