Generalizing the Dempster-Shafer Theory to Fuzzy Sets
  With the desire to apply the Dempster-Shafer theory to complex real world
problems where the evidential strength is often imprecise and vague, several
attempts have been made to generalize the theory. However, the important
concept in the D-S theory that the belief and plausibility functions are lower
and upper probabilities is no longer preserved in these generalizations. In
this paper, we describe a generalized theory of evidence where the degree of
belief in a fuzzy set is obtained by minimizing the probability of the fuzzy
set under the constraints imposed by a basic probability assignment. To
formulate the probabilistic constraint of a fuzzy focal element, we decompose
it into a set of consonant non-fuzzy focal elements. By generalizing the
compatibility relation to a possibility theory, we are able to justify our
generalization to Dempster's rule based on possibility distribution. Our
generalization not only extends the application of the D-S theory but also
illustrates a way that probability theory and fuzzy set theory can be combined
to deal with different kinds of uncertain information in AI systems.
