Divide-and-Conquer With Sequential Monte Carlo

ABSTRACT

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved subproblems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed divide-and-conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging subproblems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem. Supplementary materials including proofs and additional numerical results are available online.

KEYWORDS:

• Bayesian methods
• Graphical models
• Hierarchical models
• Particle filters

Supplementary Materials

• Appendices: The appendices contain proofs of Propositions 1 and 2 as well as additional details and results on the numerical examples. (supplementary-material.pdf)

• Code: The code used to generate the results in Sections 5.1 and 5.2 is publicly available on GitHub, at https://github.com/freli005/divide-and-conquer-smc and https://github.com/alexandrebouchard/divide-and-conquer-smc, respectively.

Acknowledgments

This research is financially supported by: the Swedish research Council (VR) via the projects Learning of complex dynamical systems (Ref no. 637-2014-466) and Probabilistic modeling of dynamical systems (Ref no. 621-2013-5524) and the Linnaeus Center CADICS; the Engineering and Physical Sciences Research Council (UK, Ref no. EP/K021672/2), and the Natural Sciences and Engineering Research Council (Canada) via a Discovery Grant.