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Proper Scoring Rules, Dominated Forecasts, and Coherence

Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
mark{at}stat.cmu.edu
teddy{at}stat.cmu.edu
kadane{at}stat.cmu.edu

The concept of coherent probabilities and conditional probabilities through a gambling argument and through a parallel argument based on a quadratic scoring rule was introduced by de Finetti (de Finetti, B. 1974. The Theory of Probability. John Wiley & Sons, New York). He showed that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules that might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while preserving the equivalence of de Finetti's two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti's fundamental theorem of probability to deal with conditional probabilities.

Key Words: Brier score; finite additivity
History: Received on November 14, 2008. Accepted on June 22, 2009.