A generalized rule for combining Gaussian belief functions

Knowledge representations using Gaussian belief functions, which extends Dempster-Shafer theory to continuous spaces of hypotheses, involve heterogeneous structures including mass density functions, hyperplanes, and assignments relations. Combination usually entails computationally prohibitive operations like intersecting hyperplanes and multiplying and integrating mass densities. Using the matrix sweep transform, Gaussian belief functions may be expressed uniformly as matrices composed of precisions, moments, or a hybrid of both with a natural semantics. However, the resulting combination rule with polynomial O(m^{3}) complexity, derived from the direct application of Dempster's rule, is limited to representations in pure precision form and requires pre-processing transformations for other formats. This paper lifts the restriction by proposing a generalized rule that combines Gaussian belief functions directly in their hybrid matrix representations, provided that matrix elements corresponding to shared variables are expressed as precision terms. The generalized rule relaxes pre-processing requisites, reduces computational complexity further from O(m^{3}) to O(m^{2}), yielding up to 95% real-world computational savings for two-variable computations and by 99.9% for moderate to large belief functions.