On Distinct Belief Functions in the Dempster-Shafer Theory

Dempster's combination rule is the centerpiece of the Dempster-Shafer (D-S) theory of belief functions. In practice, Dempster's combination rule should only be applied to combine two distinct belief functions (in the belief function literature, distinct belief functions are also called independent belief functions). So, the question arises: what constitutes distinct belief functions? We have an answer in Dempster's multi-valued functions semantics for distinct belief functions. The probability functions on the two spaces associated with the multi-valued functions should be independent. In practice, however, we don't always associate a multi-valued function with belief functions in a model. In this article, we discuss the notion of distinct belief functions in graphical models, both directed and undirected. The idea of distinct belief functions corresponds to no double-counting of non-idempotent knowledge semantics of conditional independence. Although we discuss the notion of distinct belief functions in the context of the DS theory, the discussion is valid more broadly to many uncertainty calculi, including probability theory, possibility theory, and Spohn's epistemic belief theory.