Moment Conditions for Convergence of Particle Filters with Unbounded Importance Weights

TL;DR

This paper establishes new moment conditions for particle filter importance weights that guarantee convergence of estimates even when weights are unbounded, broadening the applicability of particle filters.

Contribution

It extends existing convergence conditions by only requiring boundedness of weight moments, not the weights themselves, and demonstrates practical models where these conditions hold.

Findings

Convergence in mean square and $L^4$ is achieved under bounded second and fourth moments.

Models with point-singularities in weights can satisfy moment conditions without boundedness.

Particle filters perform well in practice even with unbounded importance weights.

Abstract

In this paper, we derive moment conditions for particle filter importance weights, which ensure that the particle filter estimates of the expectations of bounded Borel functions converge in mean square and $L^4$ sense, and that the empirical measure of the particle filter converges weakly to the true filtering measure. The result extends the previously derived conditions by not requiring the boundedness of the importance weights, but only boundedness of second or fourth order moments. We show that the boundedness of the second order moments of the weights implies the convergence of the estimates bounded functions in the mean square sense, and the $L^4$ convergence as well as the empirical measure convergence are assured by the boundedness of the fourth order moments of the weights. We also present an example class of models and importance distributions where the moment conditions hold, but the boundedness does not. The unboundedness in these models is caused by point-singularities in the weights which still leave the weight moments bounded. We show by using simulated data that the particle filter for this kind of model also performs well in practice.