Optimal Posterior E-values with Non-Convex Parameter Sets with Applications to Voting Systems
For statisticians and political pollsters, this provides a method for sequential testing in voting systems with non-convex hypotheses, though the contribution is incremental as it extends existing e-value theory.
The paper develops a theory of posterior optimal e-values for sequential testing with non-convex parameter sets, applied to voting systems (Condorcet, Borda, and Schulze). It introduces a Frank-Wolfe algorithm to compute these e-values and demonstrates efficiency gains in power and sample size on French 2022 presidential election data.
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.