Abstract
In the context of state-space models, skeleton-based smoothing algorithms rely on a backward sampling step, which by default, has a complexity (where N is the number of particles). Existing improvements in the literature are unsatisfactory: a popular rejection sampling-based approach, as we shall show, might lead to badly behaved execution time; another rejection sampler with stopping lacks complexity analysis; yet another MCMC-inspired algorithm comes with no stability guarantee. We provide several results that close these gaps. In particular, we prove a novel nonasymptotic stability theorem, thus enabling smoothing with truly linear complexity and adequate theoretical justification. We propose a general framework, which unites most skeleton-based smoothing algorithms in the literature and allows to simultaneously prove their convergence and stability, both in online and offline contexts. Furthermore, we derive, as a special case of that framework, a new coupling-based smoothing algorithm applicable to models with intractable transition densities. We elaborate practical recommendations and confirm those with numerical experiments.
Funding Statement
The first author acknowledges a CREST PhD scholarship via AMX funding.
Acknowledgments
The first author thanks the members of his Ph.D. jury (Stéphanie Allassonière, Randal Douc, Arnaud Doucet, Anthony Lee, Pierre del Moral, Christian Robert) for helpful comments on the corresponding thesis chapter. We also thank Adrien Corenflos, Samuel Duffield, the Associate Editor and the referees for comments on a preliminary version of the paper.
Citation
Hai-Dang Dau. Nicolas Chopin. "On backward smoothing algorithms." Ann. Statist. 51 (5) 2145 - 2169, October 2023. https://doi.org/10.1214/23-AOS2324
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