Belief function ($\mathrm{Bel}$)

The belief function induced by a basic probability assignment $m$ on a finite sample space $\Theta$ isIt counts evidence committed entirely inside $A$. If a random subset $S$ has probability distribution $m$, then $\mathrm{Bel}(A)=\mathbb{P}(S\subseteq A)$. This gives a concrete interpretation of the otherwise unusual summation over subsets.

For $m(\{r\})=0.6$, $m(\{r,b\})=0.4$, the singleton beliefs are $0.6$ and zero, although $\mathrm{Bel}(\{r,b\})=1$. The remaining mass is unresolved evidence, so additivity across the two singletons fails. A belief function is paired with the plausibility function $\mathrm{Pl}(A)=1-\mathrm{Bel}(A^c)$, describing evidence compatible with $A$.^[ref]