Divide-and-Conquer Fusion
Ryan S.Y. Chan, Murray Pollock, Adam M. Johansen, Gareth O. Roberts; 24(193):1−82, 2023.
Abstract
Combining multiple distributions, referred to as sub-posteriors, into a single distribution proportional to their product is a common challenge. This problem arises in various scenarios, such as distributed ‘big data’ problems or when working under multi-party privacy constraints. Many existing approaches approximate the individual sub-posteriors for practical reasons and then either find an analytical approximation or a sample approximation of the resulting posterior obtained by pooling the products of the sub-posteriors. However, the quality of the posterior approximation for these approaches is poor when the sub-posteriors deviate significantly from a narrow range of distributional form, such as being approximately Gaussian. Recently, a Fusion approach has been proposed that obtains an exact Monte Carlo approximation of the posterior, overcoming the limitations of approximate methods. Unfortunately, existing Fusion approaches have computational limitations, especially when dealing with a large number of sub-posteriors. In this paper, we generalize the theory underlying existing Fusion approaches and incorporate the resulting methodology into a recursive divide-and-conquer sequential Monte Carlo paradigm. This ultimately results in a competitive Fusion approach that can handle increasing numbers of sub-posteriors.
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