Maximum predictive power and the superposition principle

We are looking for a way of combining experimentally determined probabilities that should yield maximum predictive power. This concept is defined as permitting calculation of the accuracy of future experimental results solely on the basis of the number of runs whose data will serve as input for making the prediction. Each probability is transformed 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.