Sequential quasi Monte Carlo

Mathieu Gerber , Nicolas Chopin (1)

1 CREST - Centre de Recherche en Économie et Statistique

We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L’Ecuyer et al. (2006). The complexity of SQMC is O(N log N ), where N is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate OP (N −1/2). The only requirement to implement SQMC is the ability to write the simulation of particle xn t given xn t−1 as a deterministic function of xn t−1 and a fixed number of uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing, unbiased likelihood evaluation, and so on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain Monte Carlo) algorithm. We establish several conver- gence results. We provide numerical evidence that SQMC may significantly outperform SMC in practical scenarios

Domaines: Statistiques [math.ST]

Mathieu Gerber, Nicolas Chopin. Sequential quasi Monte Carlo. Journal of the Royal Statistical Society. Series B. Methodological, 2015, 77 (3), pp.509-579. ⟨10.1111/rssb.12104⟩. ⟨hal-01409255⟩