Coupled conditional backward sampling particle filter

We consider the coupled conditional backward sampling particle filter (CCBPF) algorithm, which is a practically implementable coupling of two conditional backward sampling particle filter (CBPF) updates with different reference trajectories. We find that the algorithm is stable, in the sense that with fixed number of particles, the coupling time in terms of iterations increases only linearly with respect to the time horizon under a general (strong mixing) condition. This result implies a convergence bound for the iterated CBPF, without requiring the number of particles to grow as a function of time horizon. This complements the earlier findings in the literature for conditional particle filters, which assume the number of particles to grow (super)linearly in terms of the time horizon. We then consider unbiased estimators of smoothing functionals using CCBPF, and also the coupled conditional particle filter without backward sampling (CCPF) as suggested by Jacob, Lindsten and Schon [arXiv:1701.02002]. In addition to our results on the CCBPF, we provide quantitative bounds on the (one-shot) coupling of CCPF, which is shown to be well-behaved with a finite time horizon and bounded potentials, when the number of particles is increased.