Abstract: We develop a (nearly) unbiased particle filtering algorithm for a specific class of continuous-time state-space models in which (a) the latent process X_t is a linear Gaussian diffusion, and (b) the observations arise from a Poisson process with intensity λ(X_t). The likelihood and the posterior probability density function of the latent process include an intractable path integral. Our algorithm relies on using Poisson estimates to approximate this integral in an unbiased manner. We show how to tune these Poisson estimates to ensure that, with large probability, all but a few of the estimates generated by the algorithm are positive. Then setting the negative estimates to zero leads to a much smaller bias than that obtained using discretization. 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 using a Born-Wolf observation model.

Key words and phrases: Continuous-time, Cox process, diffusions, hidden Markov model, particle filter, path integral, Poisson estimate, sequential Monte Carlo.