Divide-and-conquer Sequential Monte Carlo : properties and limit theorems

TY - JOUR

T1 - Divide-and-conquer Sequential Monte Carlo : properties and limit theorems

AU - Kuntz, Juan

AU - Crucinio, Francesca

AU - Johansen, Adam M.

N1 - Publisher Copyright: This research was funded, in whole or in part, by [UKRI (EPSRC), EP/T004134/1]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant’s open access conditions.

PY - 2024/2/1

Y1 - 2024/2/1

N2 - 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, L p 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.

AB - 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, L p 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.

UR - http://www.scopus.com/inward/record.url?scp=85184030060&partnerID=8YFLogxK

U2 - 10.1214/23-AAP1996

DO - 10.1214/23-AAP1996

M3 - Article

SN - 1050-5164

VL - 34

SP - 1469

EP - 1523

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 1B

ER -