Dempster-Shafer theory

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 areThe belief function counts support committed wholly to $A$; the plausibility function also counts support not excluding $A$. These are generally not additive probability measures.^[ref]

For $\Theta =\{r,b\}$, let $m(\{r\})=0.6$ and $m(\Theta )=0.4$. Then $\mathrm{Bel}(\{r\})=0.6$ and $\mathrm{Pl}(\{r\})=1$, 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.

The plausibility function associated with a finite basic probability assignment $m$ isIt includes all evidence that does not exclude $A$, whereas the belief function includes only support wholly committed to $A$. Consequently $\mathrm{Bel}(A)\le \mathrm{Pl}(A)$.

For $m(\{r\})=0.6$ and $m(\{r,b\})=0.4$, $\mathrm{Pl}(\{r\})=1$ and $\mathrm{Pl}(\{b\})=0.4$. Their sum exceeds one, so these are not the two probabilities of an ordinary binary probability distribution. In the random-subset interpretation, $\mathrm{Pl}(A)=\mathbb{P}(S\cap A\ne \varnothing )$: some possibility in $A$ remains compatible with the evidence. The gap from the belief function measures how much support remains unresolved with respect to $A$.^[ref]

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]

On a finite sample space $\Theta$, a basic probability assignment in Dempster-Shafer theory is a function on its power set satisfyingThe mass $m(B)$ is support committed to $B$ as a whole, without allocating it among the elements of $B$. Thus $m$ is a probability distribution over subsets, and generally is not an additive probability measure on $\Theta$.

For $\Theta =\{r,b\}$, let $m(\{r\})=0.6$ and $m(\Theta )=0.4$. The second mass expresses unresolved support between $r$ and $b$; it is not a claim that $b$ has probability $0.4$. This assignment induces belief function value $0.6$ and plausibility function value one for $\{r\}$. Ordinary additive probability is recovered when all nonzero masses are on singleton subsets.^[ref]