Scaling limits of sequential Monte Carlo genealogies
27th October 2017, 3:30 pm – 4:30 pm
Main Maths Building, SM3
Sequential Monte Carlo (SMC) methods constitute a broad class of numerical approximation schemes for non-linear smoothing and filtering, rare event simulation, and many other applications. In brief, an ensemble of weighted particles is used to approximate a sequence of target densities. The ensemble is evolved from one target to the next by first resampling a new generation of particles proportional to the weights of the current ensemble, and then moving and reweighting the new generation in a manner which preserves the desired target. The resampling step induces a genealogy between the generations, which has played a role in estimating mixing rates of SMC algorithms, variance of SMC estimates, as well as the memory cost of storing SMC output. The genealogy also plays a crucial role in the mixing rate of the particle Gibbs algorithm. In this talk I will introduce SMC algorithms in detail, and show that appropriately rescaled SMC genealogies converge to a well-understood process known as the Kingman coalescent under strong but standard assumptions in the infinite particle limit. This facilitates the a priori estimation of genealogical quantities, which I will demonstrate by showing that simulated genealogies match predicted results even for relatively few particles, and when the assumptions that are needed to prove convergence fail.