Towards a unified theory of imprecise probability

Coherent upper and lower probabilities, Choquet capacities of order 2, belief functions and possibility measures are amongst the most popular mathematical models for uncertainty and partial ignorance. Examples are given to show that these models are not sufficiently general to represent some common types of uncertainty. In particular, they are not sufficiently informative about expectations and conditional probabilities. Coherent lower previsions and sets of probability measures are considerably more general, but they may not be sufficiently informative for some purposes. Two other models for uncertainty, which involve partial preference orderings and sets of desirable gambles, are discussed. These are more informative and more general than the previous models, and they may provide a suitable mathematical foundation for a unified theory of imprecise probability. (C) 2000 Elsevier Science Inc. All rights reserved.