Parametric families of continuous belief functions based on generalized Gaussian random fuzzy numbers

Abstract

The theory of epistemic random fuzzy sets is a general theory of uncertainty encompassing both possibility theory and the Dempster-Shafer theory of belief functions as special cases. Within this framework, Gaussian random fuzzy numbers have recently been introduced as a practical model of uncertainty about real variables. However, the limited flexibility of this model does not allow it to represent all kinds of beliefs encountered in applications. In this paper, it is extended in two ways. First, we study one-to-one transformations of random fuzzy sets and show that such transformations commute with combination. This property allows us to define parametric families of easily combinable random fuzzy numbers and vectors on different frames based on the Gaussian model. We then go one step further by studying mixtures of random fuzzy variables, which provide a very flexible model making it possible to construct belief functions on continuous frames with arbitrary complexity. To demonstrate the applicability and practical interest of these models, two applications are studied: the elicitation of expert beliefs about numerical quantities, and generalized Bayesian inference with weak prior information represented by random fuzzy numbers.