Unbiased estimation for hidden Markov models has been recently proposed by Jacob et al (to appear), using a coupling of two conditional particle filters (CPFs). Unbiased estimation has many advantages, such as enabling the construction of asymptotically exact confidence intervals and straightforward parallelisation. In this work we propose a new coupling of two CPFs, for unbiased estimation, that uses backward sampling steps, which is an important efficiency enhancing technique in particle filtering. We show that this coupled conditional backward sampling particle filter (CCBPF) algorithm has better 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. In contrast, current coupled CPFs require the particle number to increase with the horizon length. An important corollary of our results is a new quantitative bound for the convergence rate of the popular backward sampling conditional particle filter. Previous theoretical results have not been able to demonstrate the improvement brought by backward sampling to the CPF, whereas we provide rates showing that backward sampling ensures that the CPF can remain effective with a fixed number of particles independent of the time horizon.

中文翻译：

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耦合条件后向采样粒子滤波器

Jacob 等人最近提出了隐马尔可夫模型的无偏估计（即将出现），使用两个条件粒子滤波器 (CPF) 的耦合。无偏估计具有许多优点，例如能够构建渐近精确的置信区间和直接的并行化。在这项工作中，我们提出了一种新的两个 CPF 耦合，用于无偏估计，它使用后向采样步骤，这是粒子滤波中一种重要的提高效率的技术。我们表明，这种耦合条件后向采样粒子滤波器 (CCBPF) 算法具有更好的稳定性，因为在粒子数量固定的情况下，迭代方面的耦合时间仅相对于一般（强混合）条件。相比之下，当前耦合的 CPF 要求粒子数随着地平线长度而增加。我们结果的一个重要推论是流行的后向采样条件粒子滤波器的收敛速度的新定量界限。先前的理论结果无法证明反向采样对 CPF 带来的改进，而我们提供的速率表明反向采样确保 CPF 可以在固定数量的粒子下保持有效，而不受时间范围的影响。