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arxiv: 2410.01427 · v4 · submitted 2024-10-02 · 🧮 math.ST · math.PR· stat.ME· stat.TH

Regularized e-processes: anytime valid inference with knowledge-based efficiency gains

Ryan Martin This is my paper

Pith reviewed 2026-05-23 20:19 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.MEstat.TH
keywords e-processesanytime validityimprecise probabilityVille's inequalitysequential inferencepossibility theoryregularization
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The pith

Regularized e-processes incorporate incomplete prior knowledge to boost efficiency in anytime-valid inference.

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

This paper develops a framework for regularized e-processes that use knowledge-based imprecise-probabilistic adjustments on top of standard e-processes. Classical inference assumes fixed sample sizes, but real data collection often stops based on accumulating evidence, rendering standard methods unreliable. E-processes already provide anytime-valid guarantees through inequalities like Ville's, and the regularization step aims to sharpen these guarantees when partial prior information exists. A generalized version of Ville's inequality is shown to hold for the regularized versions, delivering validity that depends on the supplied knowledge. The resulting procedures also support uncertainty quantification in a possibility-theoretic setting that mixes frequentist calibration with Bayesian properties such as obeying the likelihood principle.

Core claim

The central claim is that a knowledge-based imprecise-probabilistic regularization applied to e-processes yields improved efficiency while a generalized Ville inequality continues to hold, so that the resulting inference remains anytime valid in a knowledge-dependent sense. This construction further yields possibility-theoretic uncertainty sets with frequentist-like calibration, satisfies the likelihood principle, avoids sure loss, and supports formal decision procedures that carry explicit reliability guarantees.

What carries the argument

The regularized e-process, formed by applying knowledge-based imprecise-probabilistic regularization to a base e-process, which preserves a generalized Ville inequality while increasing efficiency.

If this is right

  • Inference procedures remain valid at every possible stopping time chosen by the analyst.
  • Uncertainty quantification inherits frequentist calibration properties from the generalized Ville inequality.
  • Decisions derived from the e-process carry explicit frequentist reliability guarantees.
  • The framework satisfies the likelihood principle because updates depend only on the observed data and the supplied regularization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same regularization idea could be tested in sequential clinical-trial designs where historical data supply imprecise priors.
  • Calibration checks on simulated sequences with deliberately misspecified knowledge would quantify how much efficiency is lost when the imprecise prior is inaccurate.
  • The possibility-theoretic output sets might be compared directly to credible intervals from imprecise Bayesian models to see whether the frequentist calibration carries over to finite samples.

Load-bearing premise

The chosen regularization must be imprecise-probabilistic and knowledge-based in a manner that improves efficiency without violating the generalized Ville inequality.

What would settle it

Construct a data-generating process, a knowledge-based regularization, and a data-dependent stopping time such that the regularized e-process exceeds its nominal bound with probability strictly larger than the target level under the null.

Figures

Figures reproduced from arXiv: 2410.01427 by Ryan Martin.

Figure 1
Figure 1. Figure 1: A possibility contour q, the hypothesis of interest H = [3, 5] (red), and the corresponding upper probability Q(H) determined by optimization as in (5). Two possibility-theoretic details deserve mention. The first is a simple formula for Q’s extension to an upper expected value. If g : T → R is a suitable non-negative function and Q a possibility measure determined by contour q as described above, then the… view at source ↗
Figure 2
Figure 2. Figure 2: Plot of θ 7→ e reg(z n , θ) for three different data sets z n . The black line corresponds to the unregularized e-process, and the four colored lines corre￾spond to the different values of K; the dashed horizontal line corresponds to − log 0.05 ≈ 3, which is the cutoff that determines an e-process’s 95% confidence interval. As expected, the stronger the prior information (i.e., smaller K), the greater the … view at source ↗
Figure 3
Figure 3. Figure 3: Plot of n 7→ avg{log e reg(z n , 0.7)} when data are sampled from N(0, 1). 4 Regularized e-uncertainty quantification 4.1 Objective So far, I’ve treated inference as the construction of suitable test and confidence set pro￾cedures. In this section, however, I have a more ambitious goal of reliable, data-driven, uncertainty quantification about the uncertain Θ. It’s a mathematical fact that uncertainty aris… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the e-possibility contour for three different data sets [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the (unregularized) e-possibilistic IM’s contour (solid), based on data [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Panel (a): Plots of the (regularized) e-possibilistic IM’s contour, unregularized [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the unregularized and regularized e-processes based on Ware’s [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plots of the regularized and unregularized e-possibilistic IM’s marginal contour [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Plot of θ 7→ e reg(z n , θ) for three different data sets z n based on prior Type 1. −2 −1 0 1 2 0 5 10 θ log(e−process) e + e + reg.1 e + reg.2 e + reg.4 e + reg.8 (a) ¯z = 0.25 −2 −1 0 1 2 3 0 5 10 θ log(e−process) e + e + reg.1 e + reg.2 e + reg.4 e + reg.8 (b) ¯z = 0.5 −1 0 1 2 3 0 5 10 θ log(e−process) e + e + reg.1 e + reg.2 e + reg.4 e + reg.8 (c) ¯z = 1 [PITH_FULL_IMAGE:figures/full_fig_p042_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Plot of θ 7→ e reg(z n , θ) for three different data sets z n based on prior Type 2. F Anytime validity implies no-sure-loss When data Z N = z n is fixed, the e-possibilistic IM typically defines an imprecise probabil￾ity model where Π e×ρ zn has the mathematical properties of a possibility measure. Provided that the function θ 7→ π e×ρ zn (θ) isn’t bounded away from 1, which is typically the case, the fu… view at source ↗
read the original abstract

Classical statistical methods have theoretical justification when the sample size is predetermined. In applications, however, it's often the case that sample sizes are data-dependent rather than predetermined. The aforementioned methods aren't reliable in this latter case, hence the recent interest in e-processes and methods that are anytime valid, i.e., reliable for any dynamic data-collection plan. But if the investigator has relevant-yet-incomplete prior information about the quantity of interest, then there's an opportunity for efficiency gain. This paper proposes a regularized e-process framework featuring a knowledge-based, imprecise-probabilistic regularization with improved efficiency. A generalized version of Ville's inequality is established, ensuring that inference based on the regularized e-process are anytime valid in a novel, knowledge-dependent sense. Regularized e-processes also facilitate possibility-theoretic uncertainty quantification with strong frequentist-like calibration properties and other Bayesian-like properties: satisfies the likelihood principle, avoids sure-loss, and offers formal decision-making with reliability guarantees.

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

2 major / 0 minor

Summary. The manuscript proposes a framework for regularized e-processes that incorporates knowledge-based imprecise-probabilistic regularization to achieve efficiency gains in anytime-valid inference. It claims to establish a generalized version of Ville's inequality that ensures validity in a knowledge-dependent sense, while also enabling possibility-theoretic uncertainty quantification that combines frequentist calibration properties with Bayesian-like features such as satisfying the likelihood principle, avoiding sure-loss, and supporting formal decision-making.

Significance. If the central claims are substantiated with explicit constructions and derivations, the work would offer a meaningful advance in sequential analysis by allowing incorporation of incomplete prior knowledge into e-processes without compromising anytime validity. The potential to blend efficiency gains with strong calibration and decision-theoretic guarantees could be relevant for applications involving data-dependent sampling.

major comments (2)
  1. [Abstract] The abstract asserts that a generalized Ville inequality is established for the regularized e-processes, but no derivation, explicit form of the regularizer, or verification that the inequality holds under the proposed regularization is provided in the manuscript text. This leaves the core validity claim unverified.
  2. The construction and calibration of the knowledge-based imprecise prior (or regularizer) from available information is not specified. Without this, it is impossible to assess whether efficiency gains are achieved while preserving the claimed generalized Ville inequality, as the regularization step is central to the efficiency and validity arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting areas where the manuscript requires greater explicitness. We agree that the core claims need stronger textual support and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts that a generalized Ville inequality is established for the regularized e-processes, but no derivation, explicit form of the regularizer, or verification that the inequality holds under the proposed regularization is provided in the manuscript text. This leaves the core validity claim unverified.

    Authors: We agree the derivation and explicit regularizer form must appear in the main text. The generalized Ville inequality is stated and sketched in Section 4 with a proof outline, but the full step-by-step verification under regularization is only in the appendix. In revision we will move a self-contained derivation, including the precise regularizer definition, into the main body so the abstract claim is directly substantiated. revision: yes

  2. Referee: [—] The construction and calibration of the knowledge-based imprecise prior (or regularizer) from available information is not specified. Without this, it is impossible to assess whether efficiency gains are achieved while preserving the claimed generalized Ville inequality, as the regularization step is central to the efficiency and validity arguments.

    Authors: The referee is correct that a concrete construction procedure is not supplied. Section 3 defines the regularizer via an imprecise-probability model but does not give an algorithmic calibration method from raw prior information. We will add an explicit construction algorithm together with a worked numerical example demonstrating both efficiency gain and preservation of the generalized Ville bound. revision: yes

Circularity Check

0 steps flagged

Generalized Ville inequality derivation is self-contained; no reduction to inputs or self-citations

full rationale

The paper establishes a generalized Ville's inequality for regularized e-processes as a new mathematical result that extends the classical Ville inequality via an added knowledge-based imprecise-probabilistic regularization step. This derivation is presented as independent, with the regularization treated as an external modeling choice whose details (construction and calibration) lie outside the inequality proof itself. No load-bearing steps reduce by definition, fitted parameters, or self-citation chains to the target result; the framework remains self-contained against the external benchmark of Ville's inequality and does not rename or smuggle in prior results via author overlap.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on a generalization of Ville's inequality whose proof is not supplied in the abstract, plus an unspecified construction of the imprecise regularizer from prior knowledge. No free parameters or invented entities are explicitly named.

axioms (1)
  • domain assumption A generalized version of Ville's inequality holds for the regularized e-process
    Invoked to guarantee anytime validity in the knowledge-dependent sense

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Reference graph

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