The conditional backward sampling particle filter (CBPF) is a powerful Markov chain Monte Carlo sampler for general state space hidden Markov model smoothing. It was proposed as an improvement over the conditional particle filter, which is known to have an $O(T^2)$ computational time complexity under a general `strong' mixing assumption, where $T$ is the time horizon. We provide the first proof that the CBPF admits an $O(T \log T)$ time complexity under strong mixing, complementing strong empirical evidence of the superiority of the CBPF in practice. In particular, the CBPF's mixing time is upper bounded by $O(\log T)$, for any sufficiently large number of particles $N$ that depends only on the mixing assumptions and not $T$. We show that an $O(\log T)$ mixing time is optimal. The proof involves the 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 thus can also be used to construct unbiased, finite variance, estimates of functionals which have arbitrary dependence on the latent state's path, with a total expected cost of $O(T \log T)$. We also investigate other couplings, and we show some of these alternatives have improved empirical behaviour.

翻译：条件向后采样粒子滤波器（CBPF）是一种针对一般状态空间隐马尔可夫模型平滑问题的强大马尔可夫链蒙特卡洛采样器。该算法被提出作为条件粒子滤波器的改进方案，已知后者在一般“强”混合假设下具有$O(T^2)$的计算时间复杂度，其中$T$为时间跨度。我们首次证明CBPF在强混合假设下可实现$O(T \log T)$的时间复杂度，这补充了CBPF在实践中优越性的强有力经验证据。特别地，当粒子数$N$足够大（仅依赖于混合假设而与$T$无关）时，CBPF的混合时间上界为$O(\log T)$。我们证明$O(\log T)$的混合时间是最优的。该证明涉及对两个CBPF的新型耦合分析，该耦合在每个时间步对两个粒子系统实施最大耦合。该耦合具有可实施性，因此也可用于构建对潜在状态路径具有任意依赖功能的函数无偏、有限方差估计，总期望成本为$O(T \log T)$。我们还研究了其他耦合方案，并展示了其中某些替代方案具有更优的实证表现。