  Particle filters are a powerful and flexible tool for performing inference on
state-space models. They involve a collection of samples evolving over time
through a combination of sampling and re-sampling steps. The re-sampling step
is necessary to ensure that weight degeneracy is avoided. In several situations
of statistical interest, it is important to be able to compare the estimates
produced by two different particle filters; consequently, being able to
efficiently couple two particle filter trajectories is often of paramount
importance. In this text, we propose several ways to do so. In particular, we
leverage ideas from the optimal transportation literature. In general, though,
computing the optimal transport map is extremely computationally expensive; to
deal with this, we introduce computationally tractable approximations to
optimal transport couplings. We demonstrate that our resulting algorithms for
coupling two particle filter trajectories often perform orders of magnitude
more efficiently than more standard approaches.
