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# Belief function

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### Belief function

1. 1. Artificial Intelligence Version 1.0 Drop me a mail: rushdecoder@yahoo.com Find me on web: http://rushdishams.googlepages.com Your group: http://groups.google.com/group/csebatchesofrushdi
2. 2. Two gambles  we bet on a head turning up when we toss a coin that is known to be fair  we bet on the outcome of a fight between the world's greatest boxer and the world's greatest wrestler. 2RS, CSE@KUET
3. 3. Why DST  If we have absolutely no information about the coin, in probability theory,  we will assume that it would be 50% head and 50% tail  we know the coin is fair, so we know for a fact that  it would be 50% head and 50% tail.  Therefore, in the two different scenarios,  we arrive at the same conclusion.  How we present total ignorance in probability theory becomes a problem. 3RS, CSE@KUET
4. 4. Why DST  In Dempster–Shafer Theory,  for the ignorance scenario,  the belief of Head and the belief of Tail would be 0.  For the fair coin scenario,  the belief of Head would be 0.5, the belief of Tail would also be 0.5. 4RS, CSE@KUET
5. 5. Formalism 5RS, CSE@KUET
6. 6. Formalism 6RS, CSE@KUET
7. 7. Formalism 7RS, CSE@KUET
8. 8. Effects of conflict (Low Conflict)  Suppose that one doctor believes a patient has  either a brain tumor — with a probability of 0.99  or meningitis — with a probability of only 0.01.  A second doctor also believes the patient has  a brain tumor — with a probability of 0.99  and believes the patient suffers from concussion — with a probability of only 0.01.  If we calculate m (brain tumor) with Dempster’s rule, we obtain m(brain tumor)=Bel (brain tumor)=1 8RS, CSE@KUET
9. 9. Effects of conflict (High Conflict)  Suppose that one doctor believes  a patient has either meningitis with a probability of 0.99  or a brain tumor with a probability of only 0.01.  A second doctor believes  the patient suffers from concussion with a probability of 0.99  and also believes the patient has a brain tumor with a probability of only 0.01.  If we calculate m (brain tumor) with Dempster’s rule, we obtain m(brain tumor)=Bel (brain tumor)=1 9RS, CSE@KUET
10. 10. Belief and Plausibility  Shafer's framework allows for belief about propositions to be represented as intervals, bounded by two values, belief (or support) and plausibility: belief ≤ plausibility. 10RS, CSE@KUET
11. 11. Belief and Plausibility  Suppose we have a belief of 0.5 and a plausibility of 0.8 for a proposition, say “the cat in the box is dead.” This means that we have evidence that allows us to state strongly that the proposition is true with a confidence of 0.5. However, the evidence contrary to that hypothesis (i.e. “the cat is alive”) only has a confidence of 0.2. The remaining mass of 0.3 (the gap between the 0.5 supporting evidence on the one hand, and the 0.2 contrary evidence on the other) is “indeterminate,” 11RS, CSE@KUET
12. 12. Belief and Plausibility 12RS, CSE@KUET
13. 13. Belief and Plausibility 13RS, CSE@KUET
14. 14. Belief and Plausibility 14RS, CSE@KUET
15. 15. Belief and Plausibility 15RS, CSE@KUET
16. 16. Belief and Plausibility 16RS, CSE@KUET
17. 17. Belief and Plausibility 17RS, CSE@KUET
18. 18. Belief and Plausibility 18RS, CSE@KUET
19. 19. Belief and Plausibility 19RS, CSE@KUET
20. 20. Belief and Plausibility 20RS, CSE@KUET
21. 21. Belief and Plausibility 21RS, CSE@KUET
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29. 29. References  Wikipedia, http://en.wikipedia.org/wiki/Belief_functi ons 29RS, CSE@KUET
30. 30. Stole few slides from  Miguel Garcia Remesal, mgremesal@fi.upm.es 30RS, CSE@KUET