belief function

Dempster-Shafer TheoryArtificial Intelligence

Version 1.0

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Two gambleswe bet on a head turning up when we toss a

coin that is known to be fairwe bet on the outcome of a fight between the

world's greatest boxer and the world's greatest wrestler.

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Why DSTIf 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.

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Why DSTIn 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.

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Formalism

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Formalism

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Formalism

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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 0.99and 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

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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

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Belief and PlausibilityShafer's framework allows for belief about

propositions to be represented as intervals, bounded by two values, belief (or support) and plausibility:

belief ≤ plausibility.

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Belief and PlausibilitySuppose 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,”

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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Belief and Plausibility

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ReferencesWikipedia,

http://en.wikipedia.org/wiki/Belief_functions

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Stole few slides fromMiguel Garcia Remesal,

mgremesal@fi.upm.es