De-biasing particle filtering for a continuous time hidden Markov model with a Cox process observation model

TL;DR

This paper introduces a nearly unbiased particle filtering algorithm for continuous-time hidden Markov models with Poisson process observations, effectively reducing bias in complex 3D tracking applications.

Contribution

The authors develop a novel particle filter that uses Poisson estimates to unbiasedly approximate intractable integrals in continuous-time models, improving bias control.

Findings

Algorithm achieves near-unbiasedness in particle filtering.

Effective in complex 3D single molecule tracking.

Reduces bias significantly compared to discretisation methods.

Abstract

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.