Weighted regret-based likelihood: a new approach to describing uncertainty
It addresses a theoretical gap in decision-making under uncertainty for researchers in AI and statistics, but it is incremental as it builds directly on prior work.
The paper tackles the problem of defining comparative likelihood for uncertainty represented by weighted sets of probability measures, generalizing probability and lower probability orderings, and provides a complete axiomatic characterization for this regret-based likelihood.
Recently, Halpern and Leung suggested representing uncertainty by a weighted set of probability measures, and suggested a way of making decisions based on this representation of uncertainty: maximizing weighted regret. Their paper does not answer an apparently simpler question: what it means, according to this representation of uncertainty, for an event E to be more likely than an event E'. In this paper, a notion of comparative likelihood when uncertainty is represented by a weighted set of probability measures is defined. It generalizes the ordering defined by probability (and by lower probability) in a natural way; a generalization of upper probability can also be defined. A complete axiomatic characterization of this notion of regret-based likelihood is given.