Justification of Logarithmic Loss via the Benefit of Side Information

We consider a natural measure of the benefit of side information: the reduction in optimal estimation risk when side information is available to the estimator. When such a measure satisfies a natural data processing property, and the source alphabet has cardinality greater than two, we show that it is uniquely characterized by the optimal estimation risk under logarithmic loss, and the corresponding measure is equal to mutual information. Further, when the source alphabet is binary, we characterize the only admissible forms the measure of predictive benefit can assume. These results unify many causality measures in the literature as instantiations of directed information, and present a natural axiomatic characterization of mutual information without requiring the sum or recursivity property.