Upper Bounds for Correlations Between Binary Outcomes and Probabilistic Predictions

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Abstract
There are many types of diverse models that yield probabilistic predictions for outcomes that are binary. We show that a correlation between the 0, 1 binary outcomes and the prediction probabilities can yield meaningful results. The upper bound for the resulting R2 turns out to be a surprisingly simple expression for a class of situations which contain a quite general assumption on the distribution of the prediction probabilities. Some implications for subjective probability estimates are also given.