Focal points and their implications for Möbius Transforms and Dempster-Shafer Theory
This work addresses computational efficiency for researchers and practitioners using Dempster-Shafer Theory in uncertainty reasoning, though it appears incremental as it builds on existing transform methods.
The paper tackles the high computational burden of Dempster-Shafer Theory by introducing focal points to simplify zeta and Möbius transforms, reducing complexity for belief representation transformations and fusion, and generalizing conjunctive decomposition of evidence.
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 of information 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 transform and its inverse, the Möbius transform, can be exactly simplified, fitting the quantity of information contained in belief functions. Beyond that, this simplification 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 Möbius transforms can be defined. We demonstrate the interest of these general results for DST, not only for the reduction in complexity 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.