CONVERGENCE OF SEQUENTIAL MONTE CARLO-BASED SAMPLING METHODS
Jonathan H. Huggins, Daniel M. Roy · arXiv (Cornell University) · 2015
Originally designed for state-space models, Sequential Monte Carlo (SMC) methods are now routinely applied in the context of general-purpose Bayesian inference. Traditional analyses of SMC algorithms have focused on their application to estimating expectations with respect to intractable distri- butions such as those arising in Bayesian analysis. However, these algorithms can also be used to obtain approximate samples from a posterior distribution of interest. We investigate the asymptotic and non-asymptotic convergence rates of SMC from this sampling viewpoint. In particular, we study the ex- pectation of the particle that SMC produces as the number of particles tends to innity. This \expected approximation is equivalent to the law of a sample drawn from the SMC approximation. We give convergence rates of the Kullback-Leibler divergence between the target and the expected approximation. Our results apply to both deterministic and adaptive resam- pling schemes. In the adaptive setting, we introduce a novel notion of eective sample size, the 1-ESS, and show that controlling this quantity ensures sta- bility of the SMC sampling algorithm. We also introduce an adaptive version of the conditional SMC proposal, which allows us to prove quantitative bounds for rates of convergence for adaptive versions of iterated conditional sequential Monte Carlo Markov chains and associated adaptive particle Gibbs samplers.