  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.
