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The filtering of a Markov diffusion process on a manifold from countingprocess observations leads to `large' changes in the conditional distributionupon an observed event, corresponding to a multiplication of the density by theintensity function of the observation process. If that distribution isrepresented by unweighted samples or particles, they need to be jointlytransformed such that they sample from the modified distribution. In previouswork, this transformation has been approximated by a translation of all theparticles by a common vector. However, such an operation is ill-defined on amanifold, and on a vector space, a constant gain can lead to a wrong estimateof the uncertainty over the hidden state. Here, taking inspiration from thefeedback particle filter (FPF), we derive an asymptotically exact filter(called ppFPF) for point process observations, whose particles evolve accordingto intrinsic (i.e. parametrization-invariant) dynamics that are composed of thedynamics of the hidden state plus additional control terms. While not sharingthe gain-times-error structure of the FPF, the optimal control terms areexpressed as solutions to partial differential equations analogous to theweighted Poisson equation for the gain of the FPF. The proposed filter cantherefore make use of existing approximation algorithms for solutions ofweighted Poisson equations.

Asymptotically exact unweighted particle filter for manifold-valued hidden states and point process observations