Twisted Particle Filters

We investigate alternative sampling laws for particle algorithms and the influence of these laws on the efficiency of particle approximations of marginal likelihoods in hidden Markov models. Amongst a broad class of candidates we characterize the essentially unique family of particle system transition kernels which is optimal with respect to an asymptotic-in-time variance growth rate criterion. The sampling structure of the algorithm defined by these optimal transitions turns out to be only quite subtly different from standard algorithms and yet the fluctuation properties of the estimates it provides are, in some ways, dramatically different. The structure of the optimal transition suggests a new class of algorithms, which we term "twisted" particle filters, and which we validate with asymptotic analysis of a more traditional nature, in the regime where the number of particles tends to infinity.