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 shows that this rate is optimal. In contrast to our result, available results to-date are extremely conservative, suggesting $O(α^N)$ time steps are needed, for some $α>1$, 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. We also establish new propagation of chaos type results using our proof techniques, discuss implications to couplings of particle filters and an application to processing out-of-sequence measurements.

翻译：我们研究了粒子滤波器的遗忘性质，其中其状态（即粒子集合）被视为一个马尔可夫链。在对粒子滤波器底层Feynman-Kac模型施加强混合假设的条件下，我们发现粒子滤波器具有指数混合性，并在$O(\log N)$的“时间”内遗忘其初始状态，其中$N$是粒子数，时间指的是粒子滤波器算法步骤数，每一步包含选择（或重采样）和变异（或预测）操作。我们给出一个例子，证明该速率是最优的。与我们的结果相反，现有结果极为保守，表明粒子滤波器需要$O(α^N)$个时间步长（对于某个$α>1$）才能遗忘其初始值。我们还研究了条件粒子滤波器（CPF），并将遗忘结果推广到该领域。我们得出类似结论：CPF具有指数混合性，并在$O(\log N)$时间内遗忘其初始状态。为支持这一分析，我们建立了CPF的新时间一致$L^p$误差估计，这些估计可能具有独立研究价值。我们还利用证明方法建立了新的混沌传播类结果，并讨论了这些结果对粒子滤波器耦合的影响以及处理失序测量的应用。