# Divide-and-Conquer Fusion

arXiv: https://arxiv.org/abs/2110.07265 · HTML: https://mlsanity.com/paper/2110.07265/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 49 references that are in the map.
- First posted on arXiv 14 Oct 2021.

## Builds on (most cited first, 11 of 11)

- [Asymptotically Exact, Embarrassingly Parallel MCMC](https://mlsanity.com/paper/1311.4780/) (2013)
- [The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method](https://mlsanity.com/paper/1510.02451/) (2015)
- [Control Variates for Stochastic Gradient MCMC](https://mlsanity.com/paper/1706.05439/) (2017)
- [Merging MCMC Subposteriors through Gaussian-Process Approximations](https://mlsanity.com/paper/1605.08576/) (2016)
- [Divide-and-Conquer with Sequential Monte Carlo](https://mlsanity.com/paper/1406.4993/) (2014)
- [Quasi-stationary Monte Carlo and the ScaLE Algorithm](https://mlsanity.com/paper/1609.03436/) (2016)
- [Bayesian Fusion: Scalable unification of distributed statistical analyses](https://mlsanity.com/paper/2102.02123/) (2021)
- [Product-form estimators: exploiting independence to scale up Monte Carlo](https://mlsanity.com/paper/2102.11575/) (2021)
- [Monte Carlo Fusion](https://mlsanity.com/paper/1901.00139/) (2019)
- [On the exact and varepsilon-strong simulation of (jump) diffusions](https://mlsanity.com/paper/1302.6964/) (2013)
- [Joining and splitting models with Markov melding](https://mlsanity.com/paper/1607.06779/) (2016)

## Built on by (most cited first, 3 of 3)

- [Federated Generalised Variational Inference: A Robust Probabilistic Federated Learning Framework](https://mlsanity.com/paper/2502.00846/) (2025)
- [Diffusion Generative Modelling for Divide-and-Conquer MCMC](https://mlsanity.com/paper/2406.11664/) (2024)
- [Measuring the right thing: justifying metrics in AI impact assessments](https://mlsanity.com/paper/2504.05007/) (2025)

## Abstract (arXiv)

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.