Weighted regret-based likelihood: a new approach to describing uncertainty

TL;DR

This paper introduces a new way to compare the likelihood of events under uncertainty represented by weighted sets of probabilities, generalizing existing probability orderings and providing an axiomatic foundation.

Contribution

It defines a novel notion of comparative likelihood based on weighted regret, extending traditional probability and upper probability concepts.

Findings

Defines a new comparative likelihood concept for weighted probability sets.

Generalizes probability and upper probability orderings.

Provides a complete axiomatic characterization of the new likelihood measure.

Abstract

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.