On (Anti)Conditional Independence in Dempster-Shafer Theory

TL;DR

This paper extends the understanding of independence in Dempster-Shafer theory by weakening conditions for graphoidal properties, including for probabilistic belief functions with singleton focal points.

Contribution

It demonstrates that the strict positivity requirement can be relaxed to non-zero singleton commonality, preserving graphoidal properties for a broader class of belief functions.

Findings

Graphoidal properties hold under weaker conditions than previously established.

Probabilistic belief functions with singleton focal points have graphoidal independence.

The results expand the applicability of independence concepts in Dempster-Shafer theory.

Abstract

This paper verifies a result of {Shenoy:94} concerning graphoidal structure of Shenoy's notion of independence for Dempster-Shafer theory of belief functions. Shenoy proved that his notion of independence has graphoidal properties for positive normal valuations. The requirement of strict positive normal valuations as prerequisite for application of graphoidal properties excludes a wide class of DS belief functions. It excludes especially so-called probabilistic belief functions. It is demonstrated that the requirement of positiveness of valuation may be weakened in that it may be required that commonality function is non-zero for singleton sets instead, and the graphoidal properties for independence of belief function variables are then preserved. This means especially that probabilistic belief functions with all singleton sets as focal points possess graphoidal properties for independence.