Unifying Practical Uncertainty Representations: I. Generalized P-Boxes

TL;DR

This paper introduces a generalized form of p-boxes that unifies various uncertainty representations, establishing their equivalences and connections with possibility distributions and random sets, thus enhancing theoretical understanding and practical handling of uncertainty.

Contribution

It defines a generalized p-boxes framework that links p-boxes, clouds, possibility distributions, and random sets, providing a unified approach to uncertainty representation.

Findings

Generalized p-boxes are equivalent to pairs of possibility distributions.

They are a special case of random sets.

They serve as a bridge between p-boxes and clouds.

Abstract

There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study.