Optimal Posterior E-values with Non-Convex Parameter Sets with Applications to Voting Systems
2026-06-29 • Information Theory
Information Theory
AI summaryⓘ
The authors developed a new method to conduct political polls that lets researchers decide early when they have enough data to make strong conclusions. They based their approach on e-values, a recent tool for sequential testing, and designed tests specifically for different voting systems, including one called the Schulze method. They also created an efficient algorithm to compute these e-values even in complex and tricky scenarios. Their method was tested both on simulated and real data from the 2022 French presidential election, showing advantages in using fewer samples while maintaining accuracy.
sequential testinge-valuesCondorcet voting systemBorda countSchulze voting methodmultivariate Bernoulli distributioncomposite hypothesisFrank-Wolfe algorithmReverse Information Projectionstatistical power
Authors
Adrienne Tuynman, Timothée Mathieu
Abstract
We are interested in conducting political polls sequentially, so that one can stop acquiring data as soon as possible while safely yielding statistically significant results. Building off e-values, which have recently become a useful tool to create sequential testing methods, we develop a theory of posterior optimal e-values. We use voting as a convenient example on which to illustrate our method. First, we design statistical tests for Condorcet and Borda voting system, and also for Schulze voting system which we are the first to tackle statistically. Then, we study the construction of optimal sequential e-values in the deceptively simple setting of multivariate Bernoulli data, with general composite null and alternative hypothesis sets $\mathcal{H}_0$ and $\mathcal{H}_1$. We give a way to compute these e-values using an efficient Frank-Wolfe algorithm, giving a pretty general way to compute Reverse Information Projections, even when $\mathcal{H}_0$ corresponds to a non-convex parameter set. Finally, we illustrate the efficiency, both in terms of power and sample size of our method. We compare with state of the art in both simulated and real data experiments, with application to French 2022 presidential election data.