Lower Previsions is an overview of, and reference guide to, the mathematics of lower previsions. Starting from first principles—acceptability—, we derive their mathematical properties and relate them to a wide range of other uncertainty models—belief functions, Choquet capacities, possibility measures—and mathematical concepts—including filters, limits, propositional logic, integration and many other constructs from functional and convex analysis. The material of the book is advanced and aimed at researchers, postgraduate students and lecturers. It will be of interest to statisticians, probabilists, mathematicians and anyone whose field of interest includes some form of uncertainty modelling, both from a practical and a theoretical point of view.
Work on this book started about 8 years ago. The idea was, at that time, to turn the most important results in Matthias’s PhD thesis, supervised by Gert, into a coherent and more or less self-contained research monograph. Our initial plan was to focus on two things: first, the relationship between natural extension and integration and second, the discussion of lower previsions defined on unbounded gambles. It soon became clear that, in order to make the book more self-contained, we needed to include much more material on lower previsions themselves. At the same time, we gathered from conversations with close colleagues that there was a definite interest in—given the perceived lack of—a comprehensive treatment of the existing theory of lower previsions. And so we decided to include, besides our own, a number of contributions from other people, amongst whom in particular are Peter Williams, Peter Walley, Sebastian Maass, Dieter Denneberg and Enrique Miranda. The present book, therefore, differs significantly from the one we started out with. While initially, the book was mostly focused on Matthias’s PhD work, in its present form, it contains much more material and both authors have contributed to it on an equal footing.
In the first part of this book, we expose and expand on the main ideas behind the theory that deals exclusively with bounded gambles. We also discuss a wide variety of special cases that may be of interest when implementing these ideas in practical problems. In doing so, we demonstrate the unifying power behind the concept of coherent lower previsions, for uncertainty modelling as well as for functional analysis. In the second part of this book, we extend the scope of the theory of lower previsions by allowing it to deal with real gambles that are not necessarily bounded. In that part, we also deal with conditioning and provide practical constructions for extending lower previsions to unbounded gambles.
We have tried to make this book as self-contained as possible. This means, amongst other things, that we have tried to at least provide an explicit formulation—if not an actual proof—of most results that we use. We have relegated to a number of appendices supporting material that did not fit nicely into the main storyline.
If you are used to a measure-theoretic approach to probability, you may initially feel somewhat lost in this book, because we do not start out with measurability at all. Indeed, the foundations of lower previsions, for arbitrary spaces, do not rely on any notion of measurability. This may come as a surprise to some people who think that using measurability is natural and should come first. Instead, our discussion of lower previsions is founded on a notion of acceptability of gambles, which has a direct behavioural interpretation. In other words, rather than posing laws of probability, we pose laws of acceptability, from which laws of probability are derived.
As often as possible, we give detailed accounts of most steps in the proofs, with explicit references to other results that are being used. This may appear to be pedantic—or even worse, condescending—to some, but we thought it better to be too specific rather than incur the risk of explaining too little: as this is not a small book, we cannot expect any reader to remember every little result we have proved or mentioned earlier.
We hope that you will enjoy reading and working with this book as much as we have enjoyed researching and writing it.