Lower Previsions

<p>This book has two main purposes. On the one hand, it provides a<br />concise and systematic development of the theory of lower previsions,<br />based on the concept of acceptability, in spirit of the work of<br />Williams and Walley. On the other hand, it also extends this theory to<br />deal with unbounded quantities, which abound in practical<br />applications.</p> <p>Following Williams, we start out with sets of acceptable gambles. From<br />those, we derive rationality criteria—avoiding sure loss and<br />coherence—and inference methods—natural extension—for<br />(unconditional) lower previsions. We then proceed to study various<br />aspects of the resulting theory, including the concept of expectation<br />(linear previsions), limits, vacuous models, classical propositional<br />logic, lower oscillations, and monotone convergence. We discuss<br />n-monotonicity for lower previsions, and relate lower previsions with<br />Choquet integration, belief functions, random sets, possibility<br />measures, various integrals, symmetry, and representation theorems<br />based on the Bishop-De Leeuw theorem.</p> <p>Next, we extend the framework of sets of acceptable gambles to consider<br />also unbounded quantities. As before, we again derive rationality<br />criteria and inference methods for lower previsions, this time also<br />allowing for conditioning. We apply this theory to construct<br />extensions of lower previsions from bounded random quantities to a<br />larger set of random quantities, based on ideas borrowed from the<br />theory of Dunford integration.</p> <p>A first step is to extend a lower prevision to random quantities that<br />are bounded on the complement of a null set (essentially bounded<br />random quantities). This extension is achieved by a natural extension<br />procedure that can be motivated by a rationality axiom stating that<br />adding null random quantities does not affect acceptability.</p> <p>In a further step, we approximate unbounded random quantities by a<br />sequences of bounded ones, and, in essence, we identify those for<br />which the induced lower prevision limit does not depend on the details<br />of the approximation. We call those random quantities ‘previsible’. We<br />study previsibility by cut sequences, and arrive at a simple<br />sufficient condition. For the 2-monotone case, we establish a Choquet<br />integral representation for the extension. For the general case, we<br />prove that the extension can always be written as an envelope of<br />Dunford integrals. We end with some examples of the theory.</p>