Maximum predictive power and the superposition principle

In quantum physics the direct observables are probabilities of events. We ask how observed probabilities must be combined to achieve what we call maximum predictive power. According to this concept the accuracy of a prediction must only depend on the number of runs whose data serve as input for the prediction. We transform each probability to an associated variable whose uncertainty interval depends only on the amount of data and strictly decreases with it. We find that for a probability which is a function of two other probabilities maximum predictive power is achieved when linearly summing their associated variables and transforming back to a probability. This recovers the quantum mechanical superposition principle.