We develop a (nearly) unbiased particle filtering algorithm for a specific class of continuous-time state-space models, such that (a) the latent process $X_t$ is a linear Gaussian diffusion; and (b) the observations arise from a Poisson process with intensity $\lambda(X_t)$. The likelihood of the posterior probability density function of the latent process includes an intractable path integral. Our algorithm relies on Poisson estimates which approximate unbiasedly this integral. We show how we can tune these Poisson estimates to ensure that, with large probability, all but a few of the estimates generated by the algorithm are positive. Then replacing the negative estimates by zero leads to a much smaller bias than what would obtain through discretisation. We quantify the probability of negative estimates for certain special cases and show that our particle filter is effectively unbiased. We apply our method to a challenging 3D single molecule tracking example with a Born and Wolf observation model.

翻译：我们为特定类别的连续时状态-空间模型开发了(近些时候)不偏向的粒子过滤算法,以便(a) 潜伏过程$X_t$是一个直线高斯扩散过程;和(b) 观测来自Poisson过程,其强度为$@lambda(X_t)$。潜伏过程的后概率密度功能的可能性包括一个棘手的路径。我们的算法依赖于Poisson估计值,该估计值近似于这个整体。我们展示了我们如何调和这些Poisson估计值,以确保在很大的概率下,除少数外,由算法产生的估计值都是正数。然后以零取代负估计值,导致的偏差远小于通过离异化获得的偏差。我们量化某些特殊情况的负估计概率,并表明我们微粒过滤器的概率是不带偏见的。我们用我们的方法用一个充满挑战的3D单分子追踪模型和生和沃尔夫观察模型来进行。