Comparing Sequential Forecasters

Consider two or more forecasters, each making a sequence of predictions for different events over time. We ask a relatively basic question: how might we compare these forecasters, either online or post-hoc, while avoiding unverifiable assumptions on how the forecasts or outcomes were generated? This work presents a novel and rigorous answer to this question. We design a sequential inference procedure for estimating the time-varying difference in forecast quality as measured by a relatively large class of proper scoring rules (bounded scores with a linear equivalent). The resulting confidence intervals are nonasymptotically valid, and can be continuously monitored to yield statistically valid comparisons at arbitrary data-dependent stopping times ("anytime-valid"); this is enabled by adapting variance-adaptive supermartingales, confidence sequences, and e-processes to our setting. Motivated by Shafer and Vovk's game-theoretic probability, our coverage guarantees are also distribution-free, in the sense that they make no distributional assumptions on the forecasts or outcomes. In contrast to a recent work by Henzi and Ziegel, our tools can sequentially test a weak null hypothesis about whether one forecaster outperforms another on average over time. We demonstrate their effectiveness by comparing forecasts on Major League Baseball (MLB) games and statistical postprocessing methods for ensemble weather forecasts.