Lower Previsions

This book has two main purposes. On the one hand, it provides a

concise and systematic development of the theory of lower previsions,

based on the concept of acceptability, in spirit of the work of

Williams and Walley. On the other hand, it also extends this theory to

deal with unbounded quantities, which abound in practical

applications.

Following Williams, we start out with sets of acceptable gambles. From

those, we derive rationality criteria---avoiding sure loss and

coherence---and inference methods---natural extension---for

(unconditional) lower previsions. We then proceed to study various

aspects of the resulting theory, including the concept of expectation

(linear previsions), limits, vacuous models, classical propositional

logic, lower oscillations, and monotone convergence. We discuss

n-monotonicity for lower previsions, and relate lower previsions with

Choquet integration, belief functions, random sets, possibility

measures, various integrals, symmetry, and representation theorems

based on the Bishop-De Leeuw theorem.

Next, we extend the framework of sets of acceptable gambles to consider

also unbounded quantities. As before, we again derive rationality

criteria and inference methods for lower previsions, this time also

allowing for conditioning. We apply this theory to construct

extensions of lower previsions from bounded random quantities to a

larger set of random quantities, based on ideas borrowed from the

theory of Dunford integration.

A first step is to extend a lower prevision to random quantities that

are bounded on the complement of a null set (essentially bounded

random quantities). This extension is achieved by a natural extension

procedure that can be motivated by a rationality axiom stating that

adding null random quantities does not affect acceptability.

In a further step, we approximate unbounded random quantities by a

sequences of bounded ones, and, in essence, we identify those for

which the induced lower prevision limit does not depend on the details

of the approximation. We call those random quantities 'previsible'. We

study previsibility by cut sequences, and arrive at a simple

sufficient condition. For the 2-monotone case, we establish a Choquet

integral representation for the extension. For the general case, we

prove that the extension can always be written as an envelope of

Dunford integrals. We end with some examples of the theory.