Uncertain probability, regular probability interval and relative proximity

This work firstly discusses two layers of meanings about uncertain information from which the uncertain probability information is naturally introduced which can generalize some other uncertain information such as interval information, intuitionistic fuzzy information and vague information. Two decision making models are presented, called relative opinion and selected opinions. To model their high dimensional counterparts, we introduce the concepts of uncertain probability matrix, regular probability interval and BUI of probability vector mainly based on some barycentric coordinates expressions. Some related properties, remarks, high dimensional cases of relative opinion and selected opinions with numerical examples are presented. Further, we technically define the concept of relative proximity for probability vector by which it is possible to define a type of unimodal function on the space of probability vectors, which otherwise is difficult to be done. Finally, some properties of relative proximity in relation to BUI of probability vector are analyzed. This work is suitable for both practitioners and theorists devoted to uncertain models.