Hybrid Probabilistic Models of Fuzzy and Rough Events

This paper discusses the relationship between probability and fuzziness based on the process of perception. As a generalization of crisp set, fuzzy set is used to model fuzzy event as proposed by Zadeh. Similarly, we may consider rough set to represent rough event in terms of probability measure. Special attention will be given to conditional probability of fuzzy event as well as conditional probability of rough event. Their several combinations of formulation and properties are examined. In the relation to evidence theory, the probability of rough event may be considered as a connecting bridge between belief-plausibility measures and the probability measures. Moreover, generalized fuzzy-rough event is introduced to generalize both fuzzy and rough events.