An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score

The classic concept of "calibrated forecasts" and its more recent refinement, "calibeating," are defined with respect to the standard quadratic scoring rule. We extend these notions to the class of $\textit{proper}$ scoring rules (for which the best forecast is the true distribution) and define $\textit{proper-calibration}$ and $\textit{proper-calibeating}$ by requiring the errors to converge to zero uniformly over all bounded proper scoring rules. We first establish that calibration always implies proper-calibration, whereas calibeating need not imply proper-calibeating. Second, we show how to guarantee proper-calibeating and proper-multicalibeating. Finally, we demonstrate the equivalence between proper-calibration and universal no regret when best replying to forecasts in decision-making under uncertainty.

This paper evaluates how reward function choice shapes the performance and behavior of LLM forecasters. We compare five proper scoring rules as training objectives for binary forecasts of resolved real-world events. Although the rules share the same theoretical incentive for truthful probability reporting, the resulting models differ in calibration, probability use, and estimated profiles of bias, information, and noise, with smaller differences in aggregate accuracy and discrimination. The Brier-trained model has the lowest observed Brier score and highest AUC-ROC, while the log-trained model has the highest observed log score and lowest calibration error. Models with similar aggregate performance also reach that performance through different combinations of bias, information, and noise. Proper scoring rules therefore need not behave interchangeably as training objectives. Reward choice may shape not only how well an LLM forecasts, but how its forecasting errors are structured.

Reinforcement learning with verifiable rewards can in principle train calibrated probabilistic forecasters, since a proper scoring rule such as the Brier score is computed from outcomes alone and is minimized in expectation by the true probability. In practice it degrades calibration, and existing remedies address epistemic uncertainty, where a model's confidence accompanies a verifiably correct or incorrect answer. We study aleatoric forecasting, where the forecast itself is the output and the label is one stochastic outcome, taking NFL in-game win probability as a testbed with the betting market as a reference. Rewarding the realized per-play outcome fails, because the single outcome is a noisy target and the policy gradient corrupts the chain of thought. We introduce a verifiable, label-free reward, a state-conditioned empirical win rate estimated from past outcomes, that removes the label noise, and we keep the gradient off the reasoning, by direct prediction or a gradient mask, so it cannot be corrupted. Trained with this reward alone, without human labels or supervised fine-tuning, a 7B model reaches the calibration of the betting market by direct prediction and is better calibrated than a zero-shot frontier model. That frontier model and a tabular estimator reach the same Brier score as this model, identifying the market's small remaining edge as live in-game information beyond their shared inputs. Masking the gradient, rather than dropping the chain of thought, preserves reasoning from which the forecast follows, which ordinary chain-of-thought training corrupts.