Convergence properties of weighted particle islands with application to the double bootstrap algorithm

TL;DR

This paper establishes convergence and stability results for weighted particle island models, particularly the double bootstrap algorithm, demonstrating its effectiveness and robustness in parallel Monte Carlo methods.

Contribution

It introduces a general convergence framework for particle island models and proves a central limit theorem for the double bootstrap algorithm with adaptive island selection.

Findings

Established a central limit theorem for the double bootstrap algorithm.

Proved long-term numerical stability and bounded asymptotic variance.

Provided a flexible theoretical framework for analyzing complex particle island operations.

Abstract

Particle island models (Verg\'e et al., 2013) 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 bootstrap 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 for a wide range of models with possibly non-compact state space.