Editing Dempster–Shafer theory (section)

===Combining beliefs===
Beliefs from different sources can be combined with various fusion operators to model specific situations of belief fusion, e.g. with [[#Dempster's rule of combination|Dempster's rule of combination]], which combines belief constraints<ref name="Jos12">{{cite journal|author1=Jøsang, A. |author2=Simon, P.|title=Dempster's Rule as Seen by Little Colored Balls|journal=Computational Intelligence|year=2012|volume=28|issue=4|pages=453–474|doi=10.1111/j.1467-8640.2012.00421.x|s2cid=5143692}}</ref> that are dictated by independent belief sources, such as in the case of combining hints<ref name="KM95">Kohlas, J., and Monney, P.A., 1995. ''[https://books.google.com/books?id=dqnwCAAAQBAJ&dq=%22A+Mathematical+Theory+of+Hints.+An+Approach+to+the+Dempster%E2%80%93Shafer+Theory+of+Evidence%22&pg=PA3 A Mathematical Theory of Hints. An Approach to the Dempster–Shafer Theory of Evidence]''. Vol. 425 in Lecture Notes in Economics and Mathematical Systems. Springer Verlag.</ref> or combining preferences.<ref name="JH12">Jøsang, A., and Hankin, R., 2012. ''Interpretation and Fusion of Hyper Opinions in Subjective Logic''. 15th International Conference on Information Fusion (FUSION)  2012. E-{{ISBN|978-0-9824438-4-2}}, IEEE.|url=https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6289948</ref> Note that the probability masses from propositions that contradict each other can be used to obtain a measure of conflict between the independent belief sources. Other situations can be modeled with different fusion operators, such as cumulative fusion of beliefs from independent sources, which can be modeled with the cumulative fusion operator.<ref name="JDR10">{{cite journal|author1=Jøsang, A. |author2=Diaz, J. |author3=Rifqi, M. |name-list-style=amp|title=Cumulative and averaging fusion of beliefs|journal=Information Fusion|year=2010|volume=11|issue=2|pages=192–200|doi=10.1016/j.inffus.2009.05.005|citeseerx=10.1.1.615.2200 |s2cid=205432025 }}</ref>

Dempster's rule of combination is sometimes interpreted as an approximate generalisation of [[Bayes' rule]]. In this interpretation the priors and conditionals need not be specified, unlike traditional Bayesian methods, which often use a symmetry (minimax error) argument to assign prior probabilities to random variables (''e.g.'' assigning 0.5 to binary values for which no information is available about which is more likely). However, any information contained in the missing priors and conditionals is not used in Dempster's rule of combination unless it can be obtained indirectly—and arguably is then available for calculation using Bayes equations.

Dempster–Shafer theory allows one to specify a degree of ignorance in this situation instead of being forced to supply prior probabilities that add to unity. This sort of situation, and whether there is a real distinction between ''[[risk]]'' and ''[[ignorance]]'', has been extensively discussed by statisticians and economists. See, for example, the contrasting views of [[Ellsberg's paradox|Daniel Ellsberg]], [[Howard Raiffa]], [[Arrovian uncertainty|Kenneth Arrow]] and [[Knightian uncertainty|Frank Knight]].{{Citation needed|reason=The contrast between these views, or some summary of the discussion, should be referenced in the citation. The wiki article links themselves do a poor job of pointing to where the disagreement is.|date=June 2014}}