 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.

Dempster–Shafer theory, also known as evidence theory or belief function theory, is a mathematical framework for modeling uncertainty and making decisions based on evidence. Developed by Arthur Dempster and Glenn Shafer in the 1960s and 1970s, this theory generalizes the concept of probability and provides a way to combine evidence from different sources.