Introduction to Imprecise Probabilities

Chapter 2Lower previsions

Enrique Miranda¹ and Gert de Cooman²

¹Department of Statistics and Operations Research, University of Oviedo, Spain

²SYSTeMS Research Group, Ghent University, Belgium

2.1 Introduction

The material on desirability in the previous chapter constitutes an excellent foundation on which to build a general theory of imprecise probabilities. One of its drawbacks, however, is that it uses a language and mathematical approach that few experts in probability will be familiar with, or even recognise: most of the existing theory of probability rather uses the language of events, (conditional) probabilities and expectations.

The present chapter aims at bridging the gap between the material on desirability and the more traditional language of probability theory, by focusing on lower and upper previsions, which have already made an appearance in Section 1.6.3, and which are direct generalizations of the probabilities and expectations we encounter in the classical theory. We intend to present the most salient facts about the theory of coherent lower previsions, which falls squarely within the subjective approach to probability. Subjective probabilities can be given a number of different interpretations, but the one we consider here is behavioural: as we shall see, a subject's probability for an event reflects his willingness to take certain actions whose outcome depends on its occurrence, such as accepting bets on or against the event at certain betting rates.

In 1975, ...