  There has recently been considerable interest in addressing the problem of
unifying distributed statistical analyses into a single coherent inference.
This problem naturally arises in a number of situations, including in big-data
settings, when working under privacy constraints, and in Bayesian model choice.
The majority of existing approaches have relied upon convenient approximations
of the distributed analyses. Although typically being computationally
efficient, and readily scaling with respect to the number of analyses being
unified, approximate approaches can have significant shortcomings -- the
quality of the inference can degrade rapidly with the number of analyses being
unified, and can be substantially biased even when unifying a small number of
analyses that do not concur. In contrast, the recent Fusion approach of Dai et
al. (2019) is a rejection sampling scheme which is readily parallelisable and
is exact (avoiding any form of approximation other than Monte Carlo error),
albeit limited in applicability to unifying a small number of low-dimensional
analyses. In this paper we introduce a practical Bayesian Fusion approach. We
extend the theory underpinning the Fusion methodology and, by embedding it
within a sequential Monte Carlo algorithm, we are able to recover the correct
target distribution. By means of extensive guidance on the implementation of
the approach, we demonstrate theoretically and empirically that Bayesian Fusion
is robust to increasing numbers of analyses, and coherently unifying analyses
which do not concur. This is achieved while being computationally competitive
with approximate schemes.
