How to Combine Probabilistic and Possibilistic (Expert) Knowledge: Uniqueness of Reconstruction in Yager’s (Product) Approach

Often, about the same real-life system, we have both measurementrelated probabilistic information expressed by a probability measure P (S) and expert-related possibilistic information expressed by a possibility measure M(S). To get the most adequate idea about the system, we must combine these two pieces of information. For this combination, R. Yager – borrowing an idea from fuzzy logic – proposed to use the simple product t-norm, i.e., to consider a set function f(S) = P (S) · M(S). A natural question is: can we uniquely reconstruct the two parts of knowledge from this function f(S)? In this paper, we prove that while in the discrete case, the reconstruction is often not unique, in the continuous case, we can always uniquely reconstruct both components P (S) and M(S) from the combined function f(S). In this sense, Yager’s combination is indeed an adequate way to combine the two parts of knowledge.