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arxiv: 2606.03600 · v1 · pith:FVOM6CJKnew · submitted 2026-06-02 · 📊 stat.ML · cs.LG

Set-Preserving Calibration from Conformal P-Values to E-Values

Nabil Alami , Jad Zakharia , Souhaib Ben Taieb This is my paper

Pith reviewed 2026-06-28 08:08 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords conformal predictione-valuesp-valuescalibrationprediction setscross-conformal predictionefficiency gains
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The pith

A set-preserving calibrator converts conformal p-values to e-values while keeping identical prediction sets.

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

The paper establishes a way to translate conformal prediction p-values into e-values without changing the resulting prediction sets. This addresses the limitation that p-values alone restrict combining evidence from multiple models or splits. Classical p-to-e methods often produce overly conservative sets, but the new calibrator avoids that. It shows gains in efficiency and enables exact coverage in cross-conformal prediction and aggregation tasks. Readers care because it makes distribution-free uncertainty quantification more flexible without extra cost in set size.

Core claim

The authors introduce a P2E calibrator that maps valid conformal p-values to e-values such that the prediction set defined by thresholding the e-value at 1/alpha is identical to the one from the p-value at alpha. They prove this property holds and demonstrate through theory and experiments that it yields smaller or equal sets compared to prior calibrators, with applications to cross-conformal prediction achieving exact 1-alpha coverage and improved efficiency in conformal aggregation.

What carries the argument

The set-preserving P2E calibrator, a function that transforms p-values derived from nonconformity scores into e-values while ensuring the superlevel sets remain unchanged.

If this is right

  • The e-value formulation permits direct application of e-value merging techniques to combine evidence from dependent conformal procedures.
  • Cross-conformal prediction variants achieve the exact 1-alpha coverage guarantee rather than the approximate 1-2alpha.
  • Conformal aggregation methods gain efficiency while maintaining validity.
  • Randomization methods for e-values can be incorporated into conformal inference without losing set preservation.

Where Pith is reading between the lines

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

  • This approach could be tested on nonconformity scores not covered in the experiments to check if set preservation generalizes.
  • It suggests potential for sequential conformal methods where evidence accumulates over time.
  • Connections to other calibration problems in multiple testing might be explored using similar preservation ideas.

Load-bearing premise

The conformal p-values are valid, which requires the data points to be exchangeable.

What would settle it

Running the calibrator on a dataset where the original conformal p-value produces a certain set size at level alpha, and checking if the e-value version produces a different set size or fails coverage.

Figures

Figures reproduced from arXiv: 2606.03600 by Jad Zakharia, Nabil Alami, Souhaib Ben Taieb.

Figure 1
Figure 1. Figure 1: Behavior of the mapping x 7→ Fn,α(x) for varying values of n and α. p → fn,α(p)/αfn,α(α) of Proposition 2.4, that we call P2E calibrators. Recall that these functions differ from standard p-to-e calibrators, as they depend on both α and n. Additionally, we show that all P2E calibrators necessarily converge to FAoN. Proposition 2.5. Let nℓ → ∞ be any sequence such that α(nℓ + 1) ∈/ N. Then, for any Fnℓ ∈ En… view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of mean prediction set length and empirical coverage obtained using different regression algorithms and different Cross-CP methods over 100 seeds, for α = 0.1 We refer to these methods as WECA, and UR-WECA, respectively. The core advantage of the WECA method lies in its ability to select data-dependent weights that are optimized to minimize some type of criterion. In our case, we select the we… view at source ↗
Figure 3
Figure 3. Figure 3: Empirical distribution of calibration scores (California Housing) for Linear Regression model. Our empirical findings are consistent with the theoretical result above. For instance, for the Linear Regression model, the California housing dataset ( [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of coverage and mean prediction set size on the Boston dataset as a function of the number of trees in the RF model, for both CCP (green) and ECCP (blue), averaged over 25 seeds. The surrounding shaded area indicates the range of mean ± standard deviation. Miscoverage level is set to α = 0.1. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

Standard conformal prediction (CP) procedures are typically formulated in terms of p-values, but reliance on p-values alone limits flexibility, for example, when combining dependent evidence across models or data splits. Recent work has explored e-value formulations for conformal inference, yet a direct connection between p- and e-value formulations in CP has been missing, especially regarding their statistical efficiency. We first identify limitations of classical p-to-e calibrators in the CP setting, showing that they are not set-preserving and can lead to overly conservative prediction sets. To address this, we propose a novel P2E calibrator that converts conformal p-values into e-values without altering the prediction set induced by the original conformal p-value. We establish both theoretically and empirically that our calibrator can yield significant efficiency gains over existing p-to-e calibrators. This e-value formulation enables principled use of recent advances in e-value merging and randomization, where we demonstrate its impact in two applications: cross-conformal prediction (CCP), whose variants typically provide only approximate $1-2\alpha$ coverage, and conformal aggregation (CA). In both cases, our e-value-based methods satisfy the desired $1-\alpha$ coverage guarantee while improving efficiency over standard baselines. More broadly, our approach expands the flexibility of CP and opens new directions for efficient, distribution-free uncertainty quantification.

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 / 2 minor

Summary. The paper introduces a novel p-to-e (P2E) calibrator for conformal p-values that is set-preserving, meaning the induced prediction sets remain identical to those from the original conformal procedure. It claims this calibrator produces valid e-values, yields efficiency gains over classical p-to-e maps, and enables exact 1-α coverage in applications such as cross-conformal prediction (CCP) and conformal aggregation (CA), where prior variants only achieve approximate coverage.

Significance. If the set-preserving construction is valid under standard exchangeability without additional score-specific restrictions, the work would meaningfully expand the toolkit for conformal inference by allowing direct use of e-value merging and randomization techniques while preserving coverage and improving efficiency. The empirical results on CCP and CA are a strength if they demonstrate gains without post-hoc tuning.

major comments (2)
  1. [Abstract and §3 (P2E calibrator construction)] Abstract and the section introducing the P2E calibrator: the set-preserving claim requires that f(p) be constant on (0,α] and on (α,1] with a jump at α to preserve {y : f(p(y)) > c} = {y : p(y) > α}. The manuscript must explicitly state which nonconformity score properties (continuity, strict monotonicity, or absence of ties) are needed for this equivalence to hold in general; the current statement that it holds 'for the scores used in the procedure' leaves the scope of the result unclear.
  2. [Theoretical section on validity (likely §4)] Theoretical results on validity and efficiency: the e-value property E[f(p)] ≤ 1 under the null must be shown without relying on fitted parameters or self-referential definitions; if the proof reduces the efficiency gain to quantities defined in terms of the same conformal scores, the claimed gains over existing calibrators need re-examination for circularity.
minor comments (2)
  1. [Notation and definitions] Clarify notation for the threshold c in the set-equivalence definition to avoid ambiguity with the original α.
  2. [Experiments on CCP and CA] The empirical sections should report the exact nonconformity scores and data splits used to allow reproduction of the efficiency comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and §3 (P2E calibrator construction)] Abstract and the section introducing the P2E calibrator: the set-preserving claim requires that f(p) be constant on (0,α] and on (α,1] with a jump at α to preserve {y : f(p(y)) > c} = {y : p(y) > α}. The manuscript must explicitly state which nonconformity score properties (continuity, strict monotonicity, or absence of ties) are needed for this equivalence to hold in general; the current statement that it holds 'for the scores used in the procedure' leaves the scope of the result unclear.

    Authors: We agree that the set-preserving property of the proposed calibrator relies on the nonconformity scores inducing p-values whose level sets align exactly with the original conformal sets. In the revised manuscript we will explicitly state in §3 (and the abstract) that the construction assumes continuous nonconformity scores with no ties, which is the standard setting ensuring distinct p-values and exact set preservation under exchangeability. This clarifies the scope without restricting the result beyond common conformal practice. revision: yes

  2. Referee: [Theoretical section on validity (likely §4)] Theoretical results on validity and efficiency: the e-value property E[f(p)] ≤ 1 under the null must be shown without relying on fitted parameters or self-referential definitions; if the proof reduces the efficiency gain to quantities defined in terms of the same conformal scores, the claimed gains over existing calibrators need re-examination for circularity.

    Authors: The e-value property E[f(p)] ≤ 1 is established directly from the uniform validity of conformal p-values under exchangeability and the explicit functional form of our calibrator; the proof in §4 uses only the marginal distribution of p and contains no fitted parameters or self-reference. Efficiency gains are shown by direct comparison of the resulting e-values to fixed classical p-to-e maps (e.g., 1/p-type calibrators) under the same scores; because the classical maps are external benchmarks, the comparison is not circular. We will add a short clarifying paragraph in §4 to emphasize this distinction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The abstract presents a direct construction of a P2E calibrator that preserves the original conformal prediction set while producing valid e-values. No equations, self-citations, or fitted parameters are shown to reduce the central claims (set-preservation or efficiency gains) to tautologies or prior author results by definition. The derivation chain relies on standard exchangeability assumptions for conformal p-values, with theoretical and empirical validation claimed independently of the construction itself. This matches the default expectation of non-circularity for a methods paper introducing a new mapping.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no free parameters, invented entities, or non-standard axioms are explicitly introduced; the work rests on the standard exchangeability assumption of conformal prediction.

axioms (1)
  • domain assumption Data points are exchangeable
    Standard assumption invoked for validity of conformal p-values in the CP setting described in the abstract.

pith-pipeline@v0.9.1-grok · 5774 in / 1221 out tokens · 18908 ms · 2026-06-28T08:08:26.294725+00:00 · methodology

discussion (0)

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