Comparing Sequential Forecasters

TL;DR

This paper develops new sequential inference methods using confidence sequences to compare forecasters' performance over time without relying on distributional assumptions, validated on real-world data.

Contribution

It introduces novel anytime-valid confidence sequences and game-theoretic frameworks for comparing forecasters' scores in a distribution-free, sequential setting.

Findings

Confidence sequences adaptively estimate score differences.

Methods are valid at arbitrary stopping times.

Empirical validation on baseball and weather forecasts.

Abstract

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, while 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.