The conditional backward sampling particle filter (CBPF) is a powerful Markov chain Monte Carlo algorithm for general state space hidden Markov model smoothing. We show that, under a general (strong mixing) condition, its mixing time is upper bounded by $O(\log T)$ where $T$ is the time horizon. The result holds for a fixed number of particles $N$ which is sufficiently large (depending on the strong mixing constants), and therefore guarantees an overall computational complexity of $O(T\log T)$ f or general hidden Markov model smoothing. We provide an example which shows that the mixing time $O(\log T)$ is optimal. Our proof relies on analysis of a novel coupling of two CBPFs, which involves a maximal coupling of two particle systems at each time instant. The coupling is implementable, and can be used to construct unbiased, finite variance estimates of functionals which have arbitrary dependence on the latent state path, with expected $O(T \log T)$ cost. We also investigate related couplings, some of which have improved empirical behaviour.

翻译：条件反向采样粒子滤波器（CBPF）是一种用于一般状态空间隐马尔可夫模型平滑的强效马尔可夫链蒙特卡洛算法。我们证明，在一般（强混合）条件下，其混合时间上限为 $O(\log T)$，其中 $T$ 为时间范围。该结果适用于固定且足够大（取决于强混合常数）的粒子数 $N$，从而保证一般隐马尔可夫模型平滑的整体计算复杂度为 $O(T\log T)$。我们通过一个示例表明 $O(\log T)$ 的混合时间是最优的。证明基于对两个CBPF的新型耦合分析，该耦合在每个时刻对两个粒子系统采用最大耦合。该耦合是可实现的，可用于构建与潜在状态路径任意相关的泛函的无偏、有限方差估计，且预期成本为 $O(T \log T)$。我们还研究了相关耦合，其中一些在经验表现上有所改进。