  Particle filters provide Monte Carlo approximations of intractable quantities
such as point-wise evaluations of the likelihood in state space models. In many
scenarios, the interest lies in the comparison of these quantities as some
parameter or input varies. To facilitate such comparisons, we introduce and
study methods to couple two particle filters in such a way that the correlation
between the two underlying particle systems is increased. The motivation stems
from the classic variance reduction technique of positively correlating two
estimators. The key challenge in constructing such a coupling stems from the
discontinuity of the resampling step of the particle filter. As our first
contribution, we consider coupled resampling algorithms. Within bootstrap
particle filters, they improve the precision of finite-difference estimators of
the score vector and boost the performance of particle marginal
Metropolis--Hastings algorithms for parameter inference. The second
contribution arises from the use of these coupled resampling schemes within
conditional particle filters, allowing for unbiased estimators of smoothing
functionals. The result is a new smoothing strategy that operates by averaging
a number of independent and unbiased estimators, which allows for 1)
straightforward parallelization and 2) the construction of accurate error
estimates. Neither of the above is possible with existing particle smoothers.
