Abstract
We provide a comprehensive characterisation of the theoretical properties of the divide-and-conquer sequential Monte Carlo (DaC-SMC) algorithm. We firmly establish it as a well-founded method by showing that it possesses the same basic properties as conventional sequential Monte Carlo (SMC) algorithms do. In particular, we derive pertinent laws of large numbers, inequalities, and central limit theorems; and we characterize the bias in the normalized estimates produced by the algorithm and argue the absence thereof in the unnormalized ones. We further consider its practical implementation and several interesting variants; obtain expressions for its globally and locally optimal intermediate targets, auxiliary measures, and proposal kernels; and show that, in comparable conditions, DaC-SMC proves more statistically efficient than its direct SMC analogue. We close the paper with a discussion of our results, open questions, and future research directions.
Funding Statement
JK and AMJ acknowledge support from the EPSRC (grant # EP/T004134/1) and the Lloyd’s Register Foundation Programme on Data-Centric Engineering at the Alan Turing Institute. FRC acknowledges support from the EPSRC and the MRC OXWASP Centre for Doctoral Training (grant # EP/L016710/1). FRC and AMJ acknowledge further support from the EPSRC (grant # EP/R034710/1).
Acknowledgments
No new data were generated or analysed during this study.
Citation
Juan Kuntz. Francesca R. Crucinio. Adam M. Johansen. "The divide-and-conquer sequential Monte Carlo algorithm: Theoretical properties and limit theorems." Ann. Appl. Probab. 34 (1B) 1469 - 1523, February 2024. https://doi.org/10.1214/23-AAP1996
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