We study the forgetting properties of the particle filter when its state - the collection of particles - is regarded as a Markov chain. Under a strong mixing assumption on the particle filter's underlying Feynman-Kac model, we find that the particle filter is exponentially mixing, and forgets its initial state in $O(\log N )$ `time', where $N$ is the number of particles and time refers to the number of particle filter algorithm steps, each comprising a selection (or resampling) and mutation (or prediction) operation. We present an example which suggests that this rate is optimal. In contrast to our result, available results to-date are extremely conservative, suggesting $O(\alpha^N)$ time steps are needed, for some $\alpha>1$, for the particle filter to forget its initialisation. We also study the conditional particle filter (CPF) and extend our forgetting result to this context. We establish a similar conclusion, namely, CPF is exponentially mixing and forgets its initial state in $O(\log N )$ time. To support this analysis, we establish new time-uniform $L^p$ error estimates for CPF, which can be of independent interest.

翻译：我们将粒子滤波器的状态（即粒子集合）视为马尔可夫链，研究其遗忘特性。在粒子滤波器底层费曼-卡克模型的强混合假设下，我们发现粒子滤波器呈指数混合，并在 $O(\log N )$ 个“时间步”内遗忘其初始状态，其中 $N$ 为粒子数，时间步指粒子滤波器算法的步骤数（每一步包含一次选择（或重采样）和一次变异（或预测）操作）。我们通过一个示例表明该速率是最优的。与我们的结果相比，现有结果极为保守，认为粒子滤波器需要 $O(\alpha^N)$ 个时间步（其中 $\alpha>1$）才能遗忘其初始状态。我们还研究了条件粒子滤波器（CPF），并将遗忘结果推广至该领域。我们得出类似结论：CPF 呈指数混合，并在 $O(\log N )$ 个时间步内遗忘其初始状态。为支持这一分析，我们建立了 CPF 新的时域一致 $L^p$ 误差估计，该结果可能具有独立研究价值。