Properties of Marginal Sequential Monte Carlo Methods

We provide a framework which admits a number of ``marginal'' sequential Monte Carlo (SMC) algorithms as particular cases -- including the marginal particle filter [Klaas et al., 2005, in: Proceedings of Uncertainty in Artificial Intelligence, pp. 308--315], , the independent particle filter [Lin et al., 2005, Journal of the American Statistical Association 100, pp. 1412--1421] and linear-cost Approximate Bayesian Computation SMC [Sisson et al., 2007, Proceedings of the National Academy of Sciences (USA) 104, pp. 1760--1765.]. We provide conditions under which such algorithms obey laws of large numbers and central limit theorems and provide some further asymptotic characterizations. Finally, it is shown that the asymptotic variance of a class of estimators associated with certain marginal SMC algorithms is never greater than that of the estimators provided by a standard SMC algorithm using the same proposal distributions.