On the connection between probability boxes and possibility measures

TL;DR

This paper investigates the relationship between possibility measures and p-boxes, establishing conditions under which p-boxes can be considered possibility measures and demonstrating how techniques for p-boxes apply to possibility measures.

Contribution

It provides necessary and sufficient conditions for p-boxes to be possibility measures and introduces a new rule of combination for independent possibility measures.

Findings

Only 0-1-valued p-boxes are possibility measures

Almost every possibility measure can be modeled by a p-box

Derived a new rule for combining independent possibility measures

Abstract

We explore the relationship between possibility measures (supremum preserving normed measures) and p-boxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating that only those p-boxes who have 0-1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes. Doing so, we arrive at a new rule of combination for possibility measures, for the independent case.