The Dempster-Shafer theory, also known as the theory of belief functions, is a generalization of the Bayesian theory of subjective probability.
Whereas the Bayesian theory requires probabilities for each question of interest, belief functions allow us to base degrees of belief for one question on probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities;
The Dempster-Shafer theory owes its name to work by A. P. Dempster (1968) and Glenn Shafer (1976), but the theory came to the attention of AI researchers in the early 1980s, when they were trying to adapt probability theory to expert systems.
Dempster-Shafer degrees of belief resemble the certainty factors in MYCIN, and this resemblance suggested that they might combine the rigor of probability theory with the flexibility of rule-based systems.
The Dempster-Shafer theory remains attractive because of its relative flexibility. 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.
To illustrate the idea of obtaining degrees of belief for one question from subjective probabilities for another, suppose I have subjective probabilities for the reliability of my friend Betty. My probability that she is reliable is 0.9, and my probability that she is unreliable is 0.1. Suppose she tells me a limb fell on my car. This statement, which must true if she is reliable, is not necessarily false if she is unreliable. So her testimony alone justifies a 0.9 degree of belief that a limb fell on my car, but only a zero degree of belief (not a 0.1 degree of belief) that no limb fell on my car. This zero does not mean that I am sure that no limb fell on my car, as a zero probability would; it merely means that Betty’s testimony gives me no reason to believe that no limb fell on my car. The 0.9 and the zero together constitute a belief function.
To illustrate Dempster’s rule for combining degrees of belief, suppose I also have a 0.9 subjective probability for the reliability of Sally, and suppose she too testifies, independently of Betty, that a limb fell on my car. The event that Betty is reliable is independent of the event that Sally is reliable, and we may multiply the probabilities of these events; the probability that both are reliable is 0.9×0.9 = 0.81, the probability that neither is reliable is 0.1×0.1 = 0.01, and the probability that at least one is reliable is 1 – 0.01 = 0.99. Since they both said that a limb fell on my car, at least of them being reliable implies that a limb did fall on my car, and hence I may assign this event a degree of belief of 0.99. Suppose, on the other hand, that Betty and Sally contradict each other—Betty says that a limb fell on my car, and Sally says no limb fell on my car.
In this case, they cannot both be right and hence cannot both be reliable—only one is reliable, or neither is reliable. The prior probabilities that only Betty is reliable, only Sally is reliable, and that neither is reliable are 0.09, 0.09, and 0.01, respectively, and the posterior probabilities (given that not both are reliable) are 9 19 , 9 19 , and 1 19 , respectively. Hence we have a 9 19 degree of belief that a limb did fall on my car (because Betty is reliable) and a 9 19 degree of belief that no limb fell on my car (because Sally is reliable).
In summary, we obtain degrees of belief for one question (Did a limb fall on my car?) from probabilities for another question (Is the witness reliable?). Dempster’s rule begins with the assumption that the questions for which we have probabilities are independent with respect to our subjective probability judgments, but this independence is only a priori; it disappears when conflict is discerned between the different items of evidence.
Implementing the Dempster-Shafer theory in a specific problem generally involves solving two related problems.
- First, we must sort the uncertainties in the problem into a priori independent items of evidence
- Second, we must carry out Dempster’s rule computationally. These two problems and their solutions are closely related.
Sorting the uncertainties into independent items leads to a structure involving items of evidence that bear on different but related questions, and this structure can be used to make computations This can be regarded as a more general approach to representing uncertainty than the Bayesian approach. The basic idea in representing uncertainty in this model is:
- Set up a confidence interval — an interval of probabilities within which the true probability lies with a certain confidence — based on the Belief B and plausibility PL provided by some evidence E for a proposition P.
- The belief brings together all the evidence that would lead us to believe in P with some certainty
- The plausibility brings together the evidence that is compatible with P and is not inconsistent with it.
- This method allows for further additions to the set of knowledge and does not assume disjoint outcomes.
