Focal points and their implications for M\"obius Transforms and Dempster-Shafer Theory

TL;DR

This paper introduces the concept of focal points to simplify zeta and M"obius transforms in Dempster-Shafer Theory, reducing computational complexity and providing new theoretical insights into belief function decompositions.

Contribution

It presents a novel notion of focal points that simplifies key transforms in DST and extends these simplifications to functions on any partially ordered set.

Findings

Simplification of zeta and M"obius transforms using focal points

Reduction in computational complexity for belief function transformations

New generalization of evidence decomposition tied to mass functions

Abstract

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 M\"obius 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 M\"obius 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.