Convergence Properties of Weighted Particle Islands with Application to the Double Bootstrap Algorithm

Pierre Del Moral 1 Eric Moulines 2 Jimmy Olsson 3 Christelle Vergé 4
1 CQFD - Quality control and dynamic reliability
INRIA Futurs, Université Bordeaux Segalen - Bordeaux 2, Université Sciences et Technologies - Bordeaux 1, CNRS - Centre National de la Recherche Scientifique : UMR5251
Abstract : Particle island models [32] provide a means of parallelization of sequential Monte Carlo methods, and in this paper we present novel convergence results for algorithms of this sort. In particular we establish a central limit theorem—as the number of islands and the common size of the islands tend jointly to infinity—of the double boot-strap algorithm with possibly adaptive selection on the island level. For this purpose we introduce a notion of archipelagos of weighted islands and find conditions under which a set of convergence properties are preserved by different operations on such archipelagos. This theory allows arbitrary compositions of these operations to be straightforwardly analyzed, providing a very flexible framework covering the double bootstrap algorithm as a special case. Finally, we establish the long-term numerical stability of the double bootstrap algorithm by bounding its asymptotic variance under weak and easily checked assumptions satisfied typically for models with non-compact state space.
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Journal articles
stochastic systems, INFORMS Applied Probability Society, 2016, 6 (2), pp.367 - 419. 〈10.1287/15-SSY190〉
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Pierre Del Moral, Eric Moulines, Jimmy Olsson, Christelle Vergé. Convergence Properties of Weighted Particle Islands with Application to the Double Bootstrap Algorithm. stochastic systems, INFORMS Applied Probability Society, 2016, 6 (2), pp.367 - 419. 〈10.1287/15-SSY190〉. 〈hal-01593885〉

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