User:Mcgree/Zadeh’s example

Zadeh's example

In the 1960s Arthur Dempster’s laid down the foundations for of what was later called the Dempster-Shafer Theory (DST). Glenn Shafers book “Mathematical Theory of Evidence” from 1976 brought the new theory into spotlight, and thus gained much importance. As a reaction on Glenn Shafers book, in 1979 Lofti Zadeh presented an example for which Bayesian inference and in particular the DST produce results usually judged unsatisfactory or counter-intuitive. Since then many authors picked up Zadeh’s example, either to criticize the DST as a whole, or as a motivation for constructing alternative approaches.

The disagreeing Experts
''Suppose that a patient is examined by two doctors, Alice and Bob. Alice’s diagnosis is that the patient has either meningitis, with probability 0.99, or brain tumor, with probability 0.01. Bob agrees with Alice that the probability of brain tumor is 0.01, but believes that it is the probability of concussion rather than meningitis that is 0.99. Applying the Dempster rule to this situation leads to the conclusion that the belief that the patient has brain tumor is 1.0 - a conclusion that is clearly counterintuitive because both Alice and Bob agree that it is highly unlikely that the patient has a brain tumor. What is even more disconcerting is that the same conclusion (i.e., Bel(brain tumor)=l) would obtain regardless of the probabilities associated with the other possible diagnoses.'' [Zadeh]

Known variables
The Probabilities provided by Alice (A) and Bob (B) are:

$$ P_A(\operatorname{meningitis})=0.99 $$

$$ P_A(\operatorname{brain tumor})=0.01 $$

$$ P_B(brain tumor)=0.01 $$

$$ P_B(concussion)=0.99 $$

Lofti Zadeh does not explicitly wrote down the values P_A(concussion) and P_B(meningitis). However, the subsequent calculation implies a common state space: \Theta = {meningitis, brain tumor, concussion, \emptyset} and, thus implicitly provides: P_A(concussion)=0.0 P_B(meningitis)=0.0