Bridging Probability and Possibility via Bayesian Theorem

Abstract

The connection between probability and possibility has been studied for decades. Several studies have intended to provide a unified interpretation for these concepts; other studies attempt to discover their transformation relationship. This paper addresses these problems using a unified perspective. By extending the viewpoint of interpreting the grade of membership as a conditional probability, we introduce the conditional probability mass function and likelihood mass function to describe randomness and fuzziness, respectively. We draw the conclusion that conditional probability is undetermined itself and can be used for describing either randomness or fuzziness, depending on how it is interpreted. A fuzzy Bayesian theorem is presented based on the fuzziness interpretation of conditional probability. Additionally, a probability-possibility conversion is derived by assigning different interpretations for conditional probability of the Bayesian theorem, which is notably similar to Klir's normalized transformation. An example of target recognition demonstrates that the fuzzy Bayesian classifier outperforms the usual Bayesian classifier.