Information fusion via the wasserstein barycenter in the space of probability measures: Direct fusion of empirical measures and Gaussian fusion with unknown correlation

In this work, a general information fusion problem is formulated as an optimisation protocol in the space of probability measures (i.e. the so-called Wasserstein metric space). The highlevel idea is to consider the data fusion result as the probability measure that is closest to a given collection of input measures in the sense that it will minimise the (weighted) Wasserstein distance between itself and the inputs. After formulating the general information fusion protocol, we consider the explicit computation of the fusion result for two special scenarios that occur frequently in practical applications. Firstly, we show how one can compute the general outcome explicitly with two Gaussian input measures (ignoring any correlation). We then examine the consistency of this result for the scenario in which the two Gaussian inputs have an unknown (but possibly non-zero) correlation. Secondly, we show how one can compute the general fusion result explicitly given two randomly sampled (discrete) empirical measures which typically have no common underlying support. Data fusion with empirical measures as input has wide applicability in applications involving Monte Carlo estimation etc.