Comparison between evidence (Dempster-Shafer) theory and Bayesian theory

Both Bayesian theory and Dempster-Shafer (evidence) theory assign non-negative weights to set of events. In Bayesian theory, the finite set of possible events denoted by \Omega, each individual event, x is assigned a non-negative weight called probability denoted by f(x). The probabilities satisfy the following properties.

f(x)\in[0,1] for all x\in\Omega
\sum f(x)=1

In Dempster-Shafer theory, the finite set called frame of discernment, \Omega comprises all possible combination of events. Therefore, any subset of the frame of discernment can be assigned a non-negative weight called mass denoted by m(x). The masses satisfy the following properties.

m(x)\in[0,1] for all x\subseteq\Omega
\sum m(x)=1 and m(\emptyset)=0

where \emptyset is the empty set.

For example, a coin of two sides has two possible outcomes (events): head and tail. A Bayesian approach assigns probability individually to head and tail as f(head)=0.5 and f(tail)=0.5. DS theory would assign mass to each of the possibilities m({\emptyset})=0, m({head})=0.45, m({tail})=0.45 and m({head, tail})=0.1. This allows DS theory to explicitly represent an ignorance or ambiguity (uncertainty) about the situation being observed. Refer [1], [2] for more information about comparison of Bayesian theory and Dempster-Shafer theory.

[1] J. C. Hoffman and R. R. Murphy, “Comparison of Bayesian and Dempster-Shafer Theory for Sensing: A Practitioner’s Approach,” in in SPIE Proc. on Neural and Stochastic Methods in Image and Signal Processing II, 1993, pp. 266–279.
[2] P. Soundappan, E. Nikolaidis, R. T. Haftka, R. Grandhi, and R. Canfield, “Comparison of evidence theory and Bayesian theory for uncertainty modeling,” Reliability Engineering & System Safety, vol. 85, no. 1–3, pp. 295–311, Jul. 2004.

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