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