Determine whether the probability calculus is necessary for all calibrated predictors

Determine whether every predictor must satisfy the probability calculus in its event argument—that is, whether every predictor must define a probability measure—to generate calibrated predictions on related sets of events.

Background

The paper derives non-negativity, normalization, additivity, and conditional-probability relationships as conditions under which calibration on some sets of events can be extended to related sets. These arguments provide a practical reason for using the probability axioms: violations can prevent calibration from carrying over and may lead to suboptimal decisions.

The derivation does not establish that the axioms are universally required of every predictor. In particular, the authors distinguish a reason for treating the probability calculus as a useful constraint from a proof that all predictors must obey it. The unresolved issue concerns the necessity, rather than merely the usefulness, of the probability axioms for calibrated prediction.