Dempster shafer

Dempster-Shafer Theory
Glenn Shafer
1

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 ofbelief may or may not have the mathematical properties of probabilities; how much they differ from probabilities will depend on how closely the two questions are related. The Dempster-Shafer theory owes its name to work by A. P. Dempster (1968) and Glenn Shafer (1976), but the kind of reasoning the theory uses can be found as far back as the seventeenth century. The theory came to the attention of AIresearchers 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. Subsequent work has made clear that the management of uncertainty inherently requires morestructure than is available in simple rule-based systems, but 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, and Dempster's rule for combining such degrees of belief when they are based on independentitems 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 limbfell 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 maymultiply the probabilities of these events; the probability that both are reliable is 0.9x0.9 = 0.81, the probability that neither is reliable is 0.1x0.1 = 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 ofbelief 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 9 9 1 9 probabilities (given that not both are reliable) are 19 , 19 , and 19 , respectively. Hence we have a 19 degree of 9 belief that a limb did fall on my car (because Betty is reliable) and a 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...