Conditional Probability

Abstract

Abstract Both the Bayesian and frequentist approaches to conditional probability are presented. The undesirable consequences of using any statistical decision procedure that is not directly based upon conditional probability are discussed. These include sure loss as in the coherency theory of de Finetti; and the uniform improvement in risk of any non‐Bayes statistical decision procedure by a computable Bayesian decision procedure. By means of the evaluation game, it is then shown how the latter translates into extremely poor performance for such a non‐Bayes procedure in repeated applications such as arise in the frequentist theory. A version of these results is then presented for prediction. In all such applications, any non‐Bayes decision procedure in common use can be replaced by a uniformly better Bayes procedure using simple linear programming methods. Finally, issues regarding finite versus infinite partitions and the countably additive approach of Kolmogorov are discussed.