Efficient M\"obius Transformations and their applications to Dempster-Shafer Theory: Clarification and implementation

TL;DR

This paper introduces Efficient M"obius Transformations (EMT), which significantly reduce computational complexity in Dempster-Shafer Theory by exploiting lattice structures and belief source information, with practical implementation details.

Contribution

It proposes EMT sequences that optimize DST computations by combining lattice structure and belief source information, improving efficiency over existing algorithms.

Findings

EMT has lower complexity than traditional lattice-based algorithms.

EMT effectively fuses belief sources in DST.

Applicability extends to any function in finite distributive lattices.

Abstract

Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a high computational burden. A lot of work has been done to reduce the complexity of computations used in information fusion with Dempster's rule. The main approaches exploit either the structure of Boolean lattices or the information contained in belief sources. Each has its merits depending on the situation. In this paper, we propose sequences of graphs for the computation of the zeta and M\"obius transformations that optimally exploit both the structure of distributive semilattices and the information contained in belief sources. We call them the Efficient M\"obius Transformations (EMT). We show that the complexity of the EMT is always inferior to the complexity of algorithms that consider the whole lattice, such as the Fast M\"obius Transform (FMT) for all DST transformations. We then explain how to use them to fuse two belief sources. More generally, our EMTs apply to any function in any finite distributive lattice, focusing on a meet-closed or join-closed subset. This article extends our work published at the international conference on Scalable Uncertainty Management (SUM). It clarifies it, brings some minor corrections and provides implementation details such as data structures and algorithms applied to DST.