Professor Shafer is developing a foundation for probability that accommodates the diversity of its uses in the normative and descriptive sciences. This foundation goes against the grain of established interpretations of probability, but it revives many older insights. It incorporates insights from pure mathematics, where the fundamental role of martingales and filtrations in the theory of stochastic processes is well recognized, as well as insights from applied statistics, where the appeal of conditional methods has conflicted with the established interpretations. The basic idea of Professor Shafer's study is that probability should be founded not on measure theory but on the classical picture of a fair game of chance, where probabilities are simultaneously fair prices, reasonable degrees of belief, and, in some cases, long-run frequencies. The applications of probability are diverse because there are many ways of relating practical problems to this classical picture. Professor Shafer's approach to probability implies a new understanding of statistical models. These models no longer represent merely frequencies or merely beliefs. Instead, they represent a relation between certain facts about the long-run and the knowledge of the observer. The project involves several different kinds of research. Alternate axiomatizations of the classical picture need to be explored and a variety of theoretical issues in statistics need to be related to the proposed new foundation. The ideas need to be related to the historical development of probability and to established debates about the meaning of probability in statistics and science. This project promises contributions to the foundations of probability important both for philosophy of science and for mathematicians interested in probability and statistics.
