Divide-and-Conquer Fusion

TL;DR

This paper introduces a recursive divide-and-conquer fusion method that improves the scalability and robustness of combining multiple sub-posteriors into a single distribution, especially in distributed and privacy-sensitive settings.

Contribution

It generalizes existing fusion theory and integrates it into a recursive SMC framework, enhancing computational efficiency and scalability for many sub-posteriors.

Findings

The new method is robust with increasing sub-posteriors.

It achieves better computational performance compared to previous fusion approaches.

The approach maintains high-quality posterior approximations.

Abstract

Combining several (sample approximations of) distributions, which we term sub-posteriors, into a single distribution proportional to their product, is a common challenge. Occurring, for instance, in distributed 'big data' problems, or when working under multi-party privacy constraints. Many existing approaches resort to approximating the individual sub-posteriors for practical necessity, then find either an analytical approximation or sample approximation of the resulting (product-pooled) posterior. The quality of the posterior approximation for these approaches is poor when the sub-posteriors fall out-with a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed which finds an exact Monte Carlo approximation of the posterior, circumventing the drawbacks of approximate approaches. Unfortunately, existing Fusion approaches have a number of computational limitations, particularly when unifying a large number of sub-posteriors. In this paper, we generalise the theory underpinning existing Fusion approaches, and embed the resulting methodology within a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately leads to a competitive Fusion approach, which is robust to increasing numbers of sub-posteriors.