Preferential Structures for Comparative Probabilistic Reasoning
This work addresses a foundational problem in AI and logic for researchers in non-monotonic logics and belief revision, offering a novel integration of qualitative and quantitative methods.
The paper tackles the discrepancy between qualitative and quantitative reasoning about relative likelihood, showing that a modified preferential approach aligns with probabilistic reasoning using sets of probability measures, enabling the use of preferential structures for comparative probabilistic reasoning with imprecise probabilities.
Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form $\varphi\succsimψ$ expressing that $\varphi$ is at least as likely as $ψ$, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative approach using probability measures. In fact, the standard preferential approach validates principles of reasoning that are incorrect from a probabilistic point of view. However, in this paper we show that a natural modification of the preferential approach yields exactly the same logical system as a probabilistic approach--not using single probability measures, but rather sets of probability measures. Thus, the same preferential structures used in the study of non-monotonic logics and belief revision may be used in the study of comparative probabilistic reasoning based on imprecise probabilities.