On (Anti)Conditional Independence in Dempster-Shafer Theory
This work is incremental, as it extends an existing result to include a broader class of belief functions in Dempster-Shafer theory, specifically benefiting researchers in uncertainty reasoning and AI.
The paper addresses the limitation in Dempster-Shafer theory where graphoidal properties for independence require strict positive normal valuations, excluding probabilistic belief functions, and shows that weakening this requirement to non-zero commonality for singleton sets preserves these properties, enabling probabilistic belief functions with singleton focal points to have graphoidal independence.
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.