  The conditional particle filter (CPF) is a promising algorithm for general
hidden Markov model smoothing. Empirical evidence suggests that the variant of
CPF with backward sampling (CBPF) performs well even with long time series.
Previous theoretical results have not been able to demonstrate the improvement
brought by backward sampling, whereas we provide rates showing that CBPF can
remain effective with a fixed number of particles independent of the time
horizon. Our result is based on analysis of a new coupling of two CBPFs, the
coupled conditional backward sampling particle filter (CCBPF). We show that
CCBPF has good stability properties 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. The
CCBPF is useful not only as a theoretical tool, but also as a practical method
that allows for unbiased estimation of smoothing expectations, following the
recent developments by Jacob et al. (to appear). Unbiased estimation has many
advantages, such as enabling the construction of asymptotically exact
confidence intervals and straightforward parallelisation.
