Editing Dempster–Shafer theory (section)

==Bayesian approximation==
The Bayesian approximation<ref>{{cite book |last1=Bauer |last2=Mathias |title=Proceedings of the Twelfth international conference on Uncertainty in artificial intelligence |date=1996 |pages=73–80}}</ref><ref>{{Cite journal|last=Voorbraak|first=Frans|date=1989-05-01|title=A computationally efficient approximation of Dempster-Shafer theory|url=https://www.sciencedirect.com/science/article/pii/S002073738980032X|journal=International Journal of Man-Machine Studies|language=en|volume=30|issue=5|pages=525–536|doi=10.1016/S0020-7373(89)80032-X|hdl=1874/26317 |issn=0020-7373|hdl-access=free}}</ref> reduces a given bpa <math>m</math> to a (discrete) probability distribution, i.e. only singleton subsets of the frame of discernment are allowed to be focal elements of the approximated version <math>\underline{m}</math> of <math>m</math>:

:<math>
\underline{m} (A) = 
	\left\{
	\begin{aligned}
		& \frac{\sum\limits_{B | A \subseteq B} m(B) }{ \sum\limits_C m(C) \cdot |C| }, &
		|A| = 1	\\
		& 0,   &  \text{otherwise}
	\end{aligned}
	\right.
</math>

It's useful for those who are only interested in the single state hypothesis.

We can perform it in the 'light' example.

{| class="wikitable"
|-
! '''Hypothesis''' !! <math>m_1</math> !! <math>m_2</math> !! <math>m_{1,2}</math> !! <math>\underline{m}_1</math> !! <math>\underline{m}_2</math> !! <math>\underline{m}_{1,2}</math> 
|-
| None || 0     || 0     || 0       || 0  ||  0      ||  0    
|-
| Red        || 0.35  || 0.11  || 0.32           ||  0.41|| 0.30 ||  0.37         
|-
| Yellow     || 0.25  || 0.21  || 0.33       ||  0.33|| 0.38 ||  0.38         
|-
| Green      || 0.15  || 0.33  || 0.24       ||  0.25        ||  0.32 ||  0.25         
|-
| Red or Yellow       ||  0.06  || 0.21  || 0.07       ||  0|| 0 ||  0             
|-
| Red or Green        ||  0.05  || 0.01  || 0.01       ||  0|| 0 ||  0           
|-
| Yellow or Green     ||  0.04  || 0.03  || 0.01         ||  0|| 0 ||  0         
|-
| Any        || 0.1   || 0.1   || 0.02       ||  0|| 0 ||  0          
|}