Fitting probability forecasting models by scoring rules and maximum likelihood

Probability forecasting models can be estimated using weighted score functions that (by definition) capture the performance of the estimated probabilities relative to arbitrary “baseline” probability assessments, such as those produced by another model, by a bookmaker or betting market, or by a human probability assessor. Maximum likelihood estimation (MLE) is interpretable as just one such method of optimum score estimation. We find that when MLE-based probabilities are themselves treated as the baseline, forecasting models estimated by optimizing any of the proven families of power and pseudospherical economic score functions yield the very same probabilities as MLE. The finding that probabilities estimated by optimum score estimation respond to MLE-baseline probabilities by mimicking them supports reliance on MLE as the default form of optimum score estimation.