Parallel Particle MCMC with Poisson Resampling

We introduce a new version of particle filter in which the number of "children" of a particle at a given time has a Poisson distribution. As a result, the number of particles is random and varies with time. An advantage of this scheme is that decendants of different particles can evolve independently. It makes easy to parallelize computations. We also show that the basic techniques of particle MCMC, namely particle independent Metropolis-Hastings, particle Gibbs Sampler and its version with backward sampling of ancestors, work under our Poisson resampling scheme. We prove that versions of these algorithms, suitably modified to our setup, preserve the target distribution on the space of trajectories of hidden Markov process.