Interval probability and its application to decision problems

Probability measures are well-defined ones that satisfy additivity. However, it is slightly tight because of its condition of additivity. Fuzzy measures that do not satisfy additivity have been proposed as the substitute measures. The only belief function involves a density function among them. In this paper, we propose two density functions by extending values of the probability functions to interval values, which do not satisfy additivity. According to the definition of interval probability functions, lower and upper probabilities are defined, respectively. Given interval probabilities by human intuition, the identification method for obtaining interval probabilities satisfying the normality condition is proposed. A combination rule and a conditional probability can be defined well. The properties of the proposed measure are clarified. Finally, a numerical example with respect to the Bayes theorem is shown.