The Fundamental Nature of the Log Loss Function

The standard loss functions used in the literature on probabilistic prediction are the log loss function and the Brier loss function; however, any proper loss function can be used for comparison of prediction algorithms. This note shows that the log loss function is most selective in that any prediction algorithm that is optimal for a given data sequence (in the sense of the algorithmic theory of randomness) under the log loss function will be optimal under any computable proper mixable loss function; on the other hand, there is a data sequence and a prediction algorithm that is optimal for that sequence under the Brier loss function but not under the log loss function.