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arxiv: 2606.29998 · v1 · pith:4P243QDPnew · submitted 2026-06-29 · 🧮 math.ST · cs.IT· math.IT· stat.ME· stat.TH

Optimal Posterior E-values with Non-Convex Parameter Sets with Applications to Voting Systems

Adrienne Tuynman , Timoth\'ee Mathieu This is my paper

Pith reviewed 2026-06-30 04:19 UTC · model grok-4.3

classification 🧮 math.ST cs.ITmath.ITstat.MEstat.TH
keywords e-valuessequential testingvoting systemsFrank-Wolfe algorithmreverse information projectionnon-convex parameter setsmultivariate Bernoulli
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The pith

Optimal posterior e-values for non-convex hypothesis sets in sequential testing can be computed using the Frank-Wolfe algorithm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops posterior optimal e-values for sequential hypothesis testing with general composite null and alternative sets in multivariate Bernoulli data. It applies the approach to design tests for Condorcet and Borda voting systems and provides the first statistical treatment of the Schulze system. The central contribution is an efficient Frank-Wolfe algorithm that solves the reverse information projection needed for these e-values even when the null parameter set is non-convex. The method is illustrated on both simulated data and French 2022 presidential election data, showing gains in power and reduced sample sizes relative to prior techniques.

Core claim

The paper establishes that posterior optimal e-values can be obtained by solving a reverse information projection problem over composite sets H0 and H1, and that this optimization can be performed efficiently with the Frank-Wolfe algorithm even when H0 is non-convex. This construction yields valid sequential tests that permit early stopping while controlling error rates for outcomes under Condorcet, Borda, and Schulze voting rules.

What carries the argument

The Frank-Wolfe algorithm applied to the reverse information projection optimization that defines posterior optimal e-values.

Load-bearing premise

The Frank-Wolfe algorithm is assumed to locate the global optimum for the e-value optimization problem despite the non-convexity of the parameter sets.

What would settle it

On a small non-convex instance where the true optimal e-value can be found by exhaustive search or convex relaxation, the Frank-Wolfe output differs from that optimum by more than numerical tolerance.

Figures

Figures reproduced from arXiv: 2606.29998 by Adrienne Tuynman, Timoth\'ee Mathieu.

Figure 1
Figure 1. Figure 1: Representation of the parameter set Φ1 for which 1 is winner, for the three voting systems we consider. 2.1.1 Condorcet When discussing preference voting, the first option that comes to mind is the Condorcet winner. The concept is simple: if an option i is preferred to any option j, then it is a Condorcet winner. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the probability of deciding “Reject [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plot of estimated Eδϕ [log(E)], with ϕ = (x, 0.6, 0.1). TG POE 4.60 4.65 4.70 4.75 G100(E, , 10 0) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of confidence 95% intervals for G100(E, π, 100) using bootstrap confidence interval (500 000 repetitions) in Borda voting system with 3 candidates, with π the Jeffrey prior restricted to Φ1. points. We plot in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the rejection probability (on the left) and the average stopping time (right) for GRO [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the rejection probability (on the left) and the average stopping time (right) for [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the sequential e-values for the candidates in the 2022 French presidential elec [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the sample complexity required to identify a single winner using three [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a theory of posterior optimal e-values for sequential hypothesis testing with composite hypotheses on multivariate Bernoulli data. It applies the framework to construct statistical tests for Condorcet, Borda, and Schulze voting systems (the latter claimed as a first statistical treatment), proposes an efficient Frank-Wolfe algorithm to compute the associated reverse information projections even when the null parameter set H0 is non-convex, and illustrates the method's efficiency via power and sample-size comparisons on simulated data and French 2022 presidential election data.

Significance. If the central computational claim holds, the work supplies a practical, general-purpose method for obtaining optimal e-values under non-convex hypothesis classes, extending e-value methodology to sequential testing problems with complex geometry. Credit is due for the explicit application to voting systems and the empirical comparisons with existing methods.

major comments (1)
  1. [Section describing the Frank-Wolfe algorithm] Section describing the Frank-Wolfe algorithm (the paragraph beginning 'We give a way to compute these e-values using an efficient Frank-Wolfe algorithm'): the claim that this procedure computes the globally optimal reverse information projection for arbitrary non-convex H0 lacks supporting analysis. Standard Frank-Wolfe convergence theory requires a convex feasible set to guarantee global optimality via the linear minimization oracle; on non-convex sets the method converges at best to stationary points. No additional structure (hidden convexity, special geometry of the objective, or custom convergence proof) is identified to restore global optimality, which is load-bearing for the headline optimality result.
minor comments (1)
  1. The abstract states that the Schulze system is tackled 'for the first time' statistically; a brief literature pointer or footnote confirming the absence of prior e-value or sequential tests would strengthen this claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Section describing the Frank-Wolfe algorithm] Section describing the Frank-Wolfe algorithm (the paragraph beginning 'We give a way to compute these e-values using an efficient Frank-Wolfe algorithm'): the claim that this procedure computes the globally optimal reverse information projection for arbitrary non-convex H0 lacks supporting analysis. Standard Frank-Wolfe convergence theory requires a convex feasible set to guarantee global optimality via the linear minimization oracle; on non-convex sets the method converges at best to stationary points. No additional structure (hidden convexity, special geometry of the objective, or custom convergence proof) is identified to restore global optimality, which is load-bearing for the headline optimality result.

    Authors: We agree that the manuscript does not supply a convergence analysis or special structure establishing global optimality of the Frank-Wolfe iterates for arbitrary non-convex H0. Standard theory indeed yields only stationarity in the non-convex case. The algorithm is offered as a practical computational tool whose performance is illustrated empirically on the voting examples; the general claim of global optimality for non-convex sets is not rigorously justified in the current text. We will revise the relevant section to qualify the claim accordingly, stating that the procedure computes a stationary point and that global optimality is not guaranteed without additional assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract and description present a computational contribution via Frank-Wolfe for reverse information projections on non-convex H0, without any quoted steps that reduce claims to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or derivations are shown that equate outputs to inputs by construction. The method is offered as an independent algorithmic tool, making the derivation self-contained per the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the multivariate Bernoulli model and Frank-Wolfe convergence are implicit background assumptions not detailed here.

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