Conditional Random-Fuzzy Variables Representing Measurement Results

Conditional probability distributions and Bayes' theorem are important and powerful tools in measurement, whenever an a priori knowledge about the measurand is available. It is well known that, thanks to Bayes' theorem, a new information about the measurand coming from a measurement result can be used to revise the a priori knowledge refining its uncertainty. Of course, this tool can be used only if both the a priori knowledge and the new information are expressed in terms of probability distributions. However, according to a recent approach to uncertainty evaluation, measurement results can be also expressed using random-fuzzy variables (RFVs), that is, using possibility distributions, instead of probability distributions. This paper proposes an extension of Bayes' theorem to the possibility domain, thus leading to the definition of conditional RFVs. A simple experimental example is also considered to prove the effectiveness of the proposed extension.