Asymptotic genealogies of interacting particle systems with an application to sequential Monte Carlo

Abstract

We study weighted particle systems in which new generations are resampled from current particles with probabilities proportional to their weights. This covers a broad class of sequential Monte Carlo (SMC) methods, widely-used in applied statistics and cognate disciplines. We consider the genealogical tree embedded into such particle systems, and identify conditions, as well as an appropriate time-scaling, under which they converge to the Kingman n-coalescent in the infinite system size limit in the sense of finite-dimensional distributions. Thus, the tractable n-coalescent can be used to predict the shape and size of SMC genealogies, as we illustrate by characterising the limiting mean and variance of the tree height. SMC genealogies are known to be connected to algorithm performance, so that our results are likely to have applications in the design of new methods as well. Our conditions for convergence are strong, but we show by simulation that they do not appear to be necessary.

Simple conditions for convergence of sequential Monte Carlo genealogies with applications

“…The result is known to apply to standard SMC with multinomial resampling [20,Corollary 1]. We additionally prove convergence for any resampling scheme based on stochastic rounding, described in Definition 4.1.…”

“…We require control over only the second and third moments of the marginal family size of each parent. This builds upon [21], a slight error in which was corrected in [20]. Ours is a substantial improvement over that work, which requires additional control over fourth moments, including cross-terms, to obtain the same convergence result.…”

“…The series of Lemmata 3.3-3.5 below show that the assumptions (3.3)-(3.5) follow from (3.2). Lemma 3.6 allows us to remove condition (3.6) by improving upon some arguments from the proof of [20,Theorem 1]; this argument is presented in detail in Section 3.1.…”

“…This is important for tuning the parameters in settings such as particle MCMC [1], fixed-lag smoothing, and stable variance estimation [28]. It also provides estimates of the memory cost associated with path storage, building on the results of [18] and [20,Corollary 2].…”

Smoothing With Couplings of Conditional Particle Filters

“…Parallelization over the N particles is mostly feasible, the main limitation coming from the resampling step (Murray et al, 2016a;Lee and Whiteley, 2015a;Whiteley et al, 2016;Paige et al, 2014;Murray et al, 2016b). The memory cost of particle filters is of order N , or N log N if trajectories are kept , see also Koskela et al (2018). Assessing the accuracy of particle approximations from a single run of these methods remains a major challenge; see Lee and Whiteley (2015b); Olsson and Douc (2017) for recent breakthroughs.…”