An Example of Dempster-Shafer Theory

The Dempster-Shafer theory is based on two ideas:

  • the idea of obtaining degrees of belief for one question from subjective probabilities for a related question.
  • Dempster’s rule for combining such degrees of belief when they are based on independent items of evidence.

In Shafer’s note, a good example is given to demonstrate the two ideas.

Degree of belief

Suppose Yang Liu comes to me and tell me a limb fell on my car. I have no idea what is the probability that a limb fells on my car. But my probability that Yang is reliable is 0.9, then my probability he is not reliable is 0.1. Let A = there is a limb on my car; B = there is no limb on my car.

  • Case 1: Yang is on the reliable part, then there is a limb falls on my car. This means degree of belief Bel(A) = 0.9.
  • Case 2: Yang is on the unreliable part, it does not mean no limb on my car, but the degree of belief Bel(B) = 0. Because his testimony gives me no reason to believe weather there is a limb on my car. This is different from the probability theory.

Dempster’s rule of combination

Suppose both Yang Liu and Peng Diao came to tell me a limb fell on my car and I have 0.9 subjective probability on both of their reliability.

  • Case 1: both of them are reliable: the probability degree of A should be 0.9\times   0.9 = 0.81.
  • Case 2: Peng is reliable while Yang is not. The probability of A should be 0.9 \times 0.1 = 0.09.
  • Case 3: Yang is reliable while Peng is not. The probability of A is also 0.09;
  • Case 4: Both Peng and Yang are not reliable, is a 0.1 \times 0.1 possibility case.

For Bel(A), we can obtain from the evidence from case 1, 2 and 3, yielding a degree of belief that Bel(A) = 0.81+0.09+0.09 = 0.99. For Bel(B), none of the case gives a reason to support, thus Bel(B)=0

What if Peng and Yang are contradict with the testimony? Suppose Yang told me there is a limb on my car, while Peng tole me there is no limb on my car. Who should I believe. Suppose I still have a 0.9 possibility on both of their reliability.

  • Case 1: Both Yang and Peng are reliable, it is a 0.81 probability case as both of them are 0.9 degree of reliability. However, it is not possible that both of them are reliable according to their testimony.
  • Case 2: Yang is reliable and Peng is not, it is a 0.09 possibility case. In this case, there is a limb on my car.
  • Case 3: Peng is reliable and Yang is not, it is also a 0.09 possibility case. In this case, there is no limb on my car.
  • Case 4: Both Peng and Yang are not reliable it is a 0.01 possibility case. Although their testimony contradicts with each other, since both of them are now on the “unreliable” status, this case could be true for A or B.

Since Case 1 is not possible, the posterior probabilities for case 2, 3, 4 are \frac{9}{19}, \frac{9}{19}, \frac{1}{19}. For Bel(A), we can obtain support from case 2, but not case 3 or 4, thus Bel(A)=\frac{9}{19}; for Bel(B), we can obtain support from case 3, not case 2 or 4, thus Bel(B) = \frac{9}{19}.

In this case, the belief of one question (Whether or not a limb falls on my car) from another (Is the whiteness reliable?).

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