We introduce a general form of sequential Monte Carlo algorithm defined in terms of a pa-rameterized resampling mechanism. We find that a suitably generalized notion of the Effective Sample Size (ESS), widely used to monitor algorithm degeneracy, appears naturally in a study of its convergence properties. We are then able to phrase sufficient conditions for time-uniform con-vergence in terms of algorithmic control of the ESS, in turn achievable by adaptively modulating the interaction between particles. This leads us to suggest novel algorithms which are, in senses to be made precise, provably stable and yet designed to avoid the degree of interaction which hinders parallelization of standard algorithms. As a byproduct we prove time-uniform convergence of the popular adaptive resampling particle filter.

We propose nested sequential Monte Carlo (NSMC), a methodology to sample from se-quences of probability distributions, even where the random variables are high-dimensional. NSMC generalises the SMC framework by re-quiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. Furthermore, NSMC can in itself be used to pro-duce such properly weighted samples. Conse-quently, one NSMC sampler can be used to con-struct an efficient high-dimensional proposal dis-tribution for another NSMC sampler, and this nesting of the algorithm can be done to an arbi-trary degree. This allows us to consider complex and high-dimensional models using SMC. We show results that motivate the efficacy of our ap-proach on several filtering problems with dimen-sions in the order of 100 to 1 000. 1.