Inference with probabilistic and fuzzy information

We adopt the interpretation of fuzzy sets in terms of coherent conditional probabilities, introduced in [2–4] and presented in this issue [7] by R. Scozzafava. Aim of this chapter is to discuss (from a syntactical point of view) which concepts of fuzzy sets theory [9] are naturally obtained simply by using coherence. In particular, we focus on operations among fuzzy subsets (and relevant t-norms and t-conorms), and on Bayesian inference procedures, when statistical and fuzzy information must be taken into account. It is obvious that in the case of inference with hybrid information the proposed interpretation of the membership provides a general and well founded framework (that of coherent conditional probability ) for merging and managing all the available information. For instance, in this frame the simplest inferential problem (to find the most probable element of a data base, starting from a probability distribution on the single elements and a fuzzy information expressed by a membership function defined on the elements of the data base) is referable to a Bayesian updating of an initial probability. The only remarkable question is that the Bayes formula is applied in an unusual semantic way: the distribution, which plays the role of “prior” probability, is here usually obtained by statistical data, whereas the membership function, which plays the role of “likelihood”, is a subjective evaluation.