Coupling of Particle Filters

Abstract

Particle filters provide Monte Carlo approximations of intractable quantities such as point-wise evaluations of the likelihood in state space models. In many scenarios, the interest lies in the comparison of these quantities as some parameter or input varies. To facilitate such comparisons, we introduce and study methods to couple two particle filters in such a way that the correlation between the two underlying particle systems is increased. The motivation stems from the classic variance reduction technique of positively correlating two estimators. The key challenge in constructing such a coupling stems from the discontinuity of the resampling step of the particle filter. As our first contribution, we consider coupled resampling algorithms. Within bootstrap particle filters, they improve the precision of finite-difference estimators of the score vector and boost the performance of particle marginal Metropolis--Hastings algorithms for parameter inference. The second contribution arises from the use of these coupled resampling schemes within conditional particle filters, allowing for unbiased estimators of smoothing functionals. The result is a new smoothing strategy that operates by averaging a number of independent and unbiased estimators, which allows for 1) straightforward parallelization and 2) the construction of accurate error estimates. Neither of the above is possible with existing particle smoothers.