Can the Various Meanings of Probability Be Reconciled

The stand-off between the frequentist and subjectivist interpretations of probability has hardened into a philosophy. According to this philosophy, probability begins as pure mathematics. The different meanings of probability correspond to different interpretations of Kolmogorov's axioms. This chapter urges a slightly different philosophy. Probability begins with the description of an unusual situation in which the different meanings of probability are unified. It is this situation—not merely the mathematics of probability—that we use in applications. And there are many ways of using it. This philosophy reconciles the various meanings of probability at a level deeper than the level of axioms. It allows us to bring together in one framework the unified eighteenth-century understanding of probability, the frequentist foundations of von Mises and Kolmogorov, and the subjectivist foundations of de Finetti. It allows us to recognize the diversity of applications of probability without positing a myriad of incompatible meanings for probability. 1 An agreement to disagree For over fifty years, there has been a consensus among philosophers, statisticians, and other probabilists about how to think about probability and its applications. According to this consensus, probability is first of all a theory in pure mathematics, based on Kolmogorov's axioms and definitions. Different interpretations of these axioms are possible, and the usefulness of each interpretation can be debated, but the mathematical theory of probability stands above the debate. As the historian Lorraine Daston puts it, “The mathematical theory itself preserves full conceptual independence from these interpretations, 1 To appear in Methodological and Quantitative Issues in the Analysis of Psychological Data, Second Edition, edited by Gideon Keren and Charles Lewis, and published by Lawrence Erlbaum, Hillsdale, New Jersey. 2 Ronald G. Harper Distinguished Professor of Business, School of Business, Summerfield Hall, University of Kansas, Lawrence, Kansas 66045. Research for this article has been partially supported by the National Science Foundation through grant IRI8902444 to the University of Kansas and grant BNS8700864 to the Center for Advanced Study in the Behavioral Sciences. The author has benefited from conversations with Robert Fogelin, David Israel, Ali Jenzarli, Don Ylvisaker, and Joe VanZandt.