Bayesian Fusion of Empirical Distributions Based on Local Density Reconstruction

Fusing two random vectors is simple, when they are characterized by continuous probability functions.
According to Bayes‘ law, fusion then consists of multiplying the two densities. When only empirical distributions
are given and a resulting empirical distribution is desired, Bayes‘ law is no longer applicable. Obviously,
fusion could now be performed by reconstructing the underlying continuous densities, subsequent multiplication,
and sampling of the result. As this is overly complicated, our goal is to perform a direct Bayesian fusion of the
two given empirical distributions. We devise a generalized multiplication procedure that mutually reweights appropriate
points of one density by local density values of the other density. The density values are efficiently estimated locally
by nearest neighbor operations. The method is symmetric in the sense that it uses points from both densities.