Log Score

Proposed by I.J. Good in 1952, the logarithmic scoring rule rewards forecasters proportionally to the log of the probability they placed on the realized outcome. If you assign 90% to an event and it occurs, your score is log(0.9), approximately -0.046 (in base 2, roughly -0.15 bits). If you assign 10% and it occurs, your score is log(0.1), approximately -2.3 nats, a much larger penalty. This structure makes the log score highly sensitive to confident wrong predictions, which is desirable when Calibration over rare events is important. It is a local scoring rule: only the probability on the realized outcome matters, not the distribution across other outcomes. This contrasts with the Brier Score, which considers the entire probability assignment.

Both the log score and Brier Score are strictly proper, meaning a forecaster cannot improve their expected score by misreporting true beliefs. The key practical difference is sensitivity: the log score approaches negative infinity as confident wrong forecasts approach 1, while the Brier score is bounded between 0 and 1. This makes the log score better suited for evaluating forecasts on low-probability events where overconfidence is most dangerous. In prediction markets, understanding log-score incentives helps traders recognize why implied probabilities should reflect true beliefs rather than rounded guesses. Platforms using log-score leaderboards, such as Metaculus at certain scoring periods, incentivize more careful probability reporting near the extremes than crowd-averaging alone would achieve.