Combining Stochastic Uncertainty and Linguistic Inexactness: Theory and Experimental Evaluation of Four Fuzzy Probability Models

Two major sources of imprecision in human knowledge, linguistic inexactness and stochastic uncertainty, are identified in this study. It is argued that since in most realistic situations these two types exist simultaneously, it is necessary to combine them in a formal framework to yield realistic solutions. This study presents such a framework by combining concepts from probability and fuzzy set theories. In this framework four models (Kwakernaak, 1978; Yager, 1979; 1984b; Zadeh, 1968; 1975) that attempt to account for the numeric or linguistic responses in various probability elicitation tasks were tested. The linguistic models were relatively effective in predicting subjects' responses compared to a random choice model. The numeric model (Zadeh, 1968) proved to be insufficient. These results and others suggest that subjects are unable to represent the full complexity of a problem. Instead they adopt a simplified view of the problem by representing vague linguistic concepts by multiple-crisp representations (the a-level sets). All of the mental computation is done at these surrogate levels.