Belief functions induced by random fuzzy sets: Application to statistical inference

It is based on the representation of elementary pieces of evidence by belief functions (defined as completely monotone set functions) and on their combination by an operator called the product-intersection rule, or Dempster's rule of combination. A belief function can be constructed by comparing a piece evidence to a scale of canonical examples such as randomly coded messages, whose meanings are determined by chance [40]. A belief function on a set Θ can be seen as being induced by a multi-valued mapping from a probability space to Ω; it is mathematically equivalent to a random set [5, 34]. As rational beliefs are essentially determined by evidence, the Dempster-Shafer (DS) theory can be regarded as a general framework for reasoning with uncertainty [11]. Shortly after the introduction of DS theory, Zadeh independently proposed another formalism, called Possibility Theory [54], in which the concept of "fuzzy restriction" plays a