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Abstract:
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Probability theory began with betting games. Beginning with Jacob Bernoulli, statisticians have suppressed the role of betting in order to make probabilities appear objective. The personalist re-interpretation of Bayesian inference has brought back one element of a betting game: a player who offers and updates betting odds. We can further enrich probability and statistics by considering an opponent who chooses which offers to take. The opponent can test the first player’s odds by trying to multiply the money he risks. Depending on circumstances and purposes, the statistician can take either role, that of Player I, who offers odds, or that of Player II, who tests them. As Player I, the statistician may think of herself as a Bayesian or simply be trying to make good probabilistic forecasts, like any other weather forecaster, political pundit, or financial analyst. As Player II, the statistician may be trying to check on Player I’s sagacity or on whether he is “calibrated” or has good “frequency properties”. Often the statistician plays both roles, constructing prediction models and then checking them against data. She is both “Bayesian” and “frequentist”.
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