Introduction to Imprecise Probabilities

Introduction

One of the big challenges for science is coping with uncertainty, omnipresent in modern societies and of ever increasing complexity. Quantitative modelling of uncertainty is traditionally based on the use of precise probabilities: for each event , a single (classical, precise) probability is assigned, typically implicitly assumed to satisfy Kolmogorov's axioms. Although there have been many successful applications of this concept, an increasing number of researchers in different areas keep warning that the concept of classical probability has severe limitations. The mathematical formalism of classical probability indispensably requires, and implicitly presupposes, an often unrealistically high level of precision and internal consistency of the information modelled, and thus relying on classical probability under complex uncertainty may lead to unjustified, and possibly deceptive, conclusions.

Against this background, a novel, more flexible theory of uncertainty has evolved: imprecise probabilities. Imprecise probabilities have proven a powerful and elegant framework for quantifying, as well as making inferences and decisions, under uncertainty. They encompass and extend the traditional concepts and methods of probability and statistics by allowing for incompleteness, ...