Abstract: Suppose we can readily access samples from but we wish to obtain samples from . The so-called Bayesian Fusion problem comes up within various areas of modern Bayesian Machine Learning, for example in the context of big data or privacy constraints, as well as more traditional statistical areas such as meta-analysis. Many approximate solutions to this problem have been proposed. However this talk will present an exact solution based on rejection sampling in an extended state space, where the accept/reject decision is carried out by simulating the skeleton of a suitably constructed auxiliary collection of Brownian bridges.