Focal points and their implications for Möbius transforms and Dempster-Shafer Theory

Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a much higher computational burden. A lot of work has been done to reduce the time complexity information fusion with Dempster's rule, which is pointwise multiplication two zeta transforms, and optimal general algorithms have found get complete definition these transforms. Yet, it shown in this paper that transform its inverse, M\"obius transform, can be exactly simplified, fitting quantity contained belief functions. Beyond that, simplification actually works for any function on partially ordered set. It relies new notion we call focal point constitutes smallest domain both transforms defined. We demonstrate interest results DST, not only reduction most transformations between representations their fusion, also theoretical purposes. Indeed, provide generalization conjunctive decomposition evidence formulas uncovering how each weight tied corresponding mass function.