Comparing Sequential Forecasters

Consider two forecasters, each making a single prediction for a sequence of events over time. We ask a relatively basic question: how might we compare these forecasters, either online or post hoc, avoiding unverifiable assumptions on how the forecasts and outcomes were generated? In this paper, we present a rigorous answer to this question by designing novel sequential inference procedures for estimating the time-varying difference in forecast scores. To do this, we employ confidence sequences (CS), which are sequences of confidence intervals that can be continuously monitored and are valid at arbitrary data dependent stopping times ("anytime-valid"). The widths of our CSs are adaptive to the underlying variance of the score differences. Underlying their construction is a game theoretic statistical framework in which we further identify e-processes and p-processes for sequentially testing a weak null hypothesis-whether one forecaster outperforms another on average (rather than always). Our methods do not make distributional assumptions on the forecasts or outcomes; our main theorems apply to any bounded scores, and we later provide alternative methods for unbounded scores. We empirically validate our approaches by comparing real-world baseball and weather forecasters.