= Dempster-Shafer theory
{c}
<Dempster-Shafer theory> represents evidence that may support a set of possibilities without assigning its support to individual possibilities. On a finite <sample space> $\Theta$, a <basic probability assignment> is a <function> $m:2^\Theta\to[0,1]$ with $m(\varnothing)=0$ and $\sum_{B\subseteq\Theta}m(B)=1$. The associated <belief function> and <plausibility function> are
$$
\operatorname{Bel}(A)=\sum_{B\subseteq A}m(B),\qquad
\operatorname{Pl}(A)=\sum_{B\cap A\ne\varnothing}m(B)=1-\operatorname{Bel}(A^c).
$$
The <belief function> counts support committed wholly to $A$; the <plausibility function> also counts support not excluding $A$. These are generally not additive <probability measures>.<Belief Functions: Introduction by Glenn Shafer>[Sections 1–2, degrees of belief and uncommitted evidence]{ref}
For $\Theta=\{r,b\}$, let $m(\{r\})=0.6$ and $m(\Theta)=0.4$. Then $\operatorname{Bel}(\{r\})=0.6$ and $\operatorname{Pl}(\{r\})=1$, while the corresponding values for $\{b\}$ are $0$ and $0.4$. The uncommitted $0.4$ has not been asserted to be evidence specifically for $b$. Rules for combining separate bodies of evidence need assumptions about how the evidence was obtained; they are not automatic consequences of these definitions.
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