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.

Funding: A. Ramdas acknowledges funding from the National Science Foundation Division of Mathematical Sciences [Grant 1916320]. Research reported in this paper was sponsored in part by the DEVCOM Army Research Laboratory under Cooperative Agreement W911NF-17-2-0196 (ARL IoBT CRA).

Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2021.0792.