On the connection between probability boxes and possibility measures

Abstract

We explore the relationship between p-boxes on totally preordered spaces and possibility measures. 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.

Extreme Points of the Core of Possibility Measures and Maxitive p-Boxes

“…The connection between univariate pboxes and possibility measures is claried in the following proposition: Proposition 3. 26 Let (F , F ) be a univariate p-box on X . Its associated upper probability is a possibility measure if and only if F or F is vacuous, meaning that…”