ST-BCP: Tightening Coverage Bound for Backward Conformal Prediction via Non-Conformity Score Transformation

Hao Zeng; Hongxin Wei; Junxian Liu

arxiv: 2602.01733 · v2 · pith:UU4UAONAnew · submitted 2026-02-02 · 📊 stat.ML · cs.LG

ST-BCP: Tightening Coverage Bound for Backward Conformal Prediction via Non-Conformity Score Transformation

Junxian Liu , Hao Zeng , Hongxin Wei This is my paper

Pith reviewed 2026-05-21 14:10 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords conformal predictionbackward conformal predictionnonconformity scorescoverage boundsMarkov's inequalityprediction setsuncertainty quantification

0 comments

The pith

A data-dependent transformation of nonconformity scores tightens the coverage bound in backward conformal prediction.

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

Backward conformal prediction controls prediction set sizes by estimating coverage guarantees with Markov's inequality. This approach often produces loose bounds, creating a noticeable gap with observed coverage rates. The paper proposes a computable data-dependent transformation of the nonconformity scores that narrows this gap. It proves the transformation improves on the basic identity method. A reader would care because it enables more precise uncertainty estimates when set sizes are capped in advance.

Core claim

In backward conformal prediction, Markov's inequality provides a coverage bound but with looseness. ST-BCP introduces a data-dependent transformation of nonconformity scores. The authors develop a specific computable transformation and prove it outperforms the identity transformation. Experiments confirm a reduction in the average coverage gap from 4.20% to 1.12%.

What carries the argument

ST-BCP's data-dependent nonconformity score transformation that tightens the Markov inequality-based coverage bound.

If this is right

The transformation yields a tighter coverage estimate than the identity transformation.
Controlled-size prediction sets come with improved accuracy in their coverage guarantees.
The validity of the conformal prediction framework remains intact under the transformation.
Benchmark tests show consistent reduction in coverage gaps across datasets.

Where Pith is reading between the lines

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

This method could improve reliability in applications requiring fixed prediction set sizes, such as resource-constrained systems.
It may inspire similar transformations in other conformal prediction variants that use bounding inequalities.
Testing on non-exchangeable data could reveal limits of the approach.

Load-bearing premise

The data-dependent transformation preserves the validity of the coverage bound derived from Markov's inequality without introducing additional bias or violating the exchangeability assumption.

What would settle it

Reproducing the experiments on common benchmarks and finding that the coverage gap does not decrease or that actual coverage falls below the estimated bound would falsify the central claim.

Figures

Figures reproduced from arXiv: 2602.01733 by Hao Zeng, Hongxin Wei, Junxian Liu.

**Figure 1.** Figure 1: Performance comparison between BCP (h = s) and our method ST-BCP (h = Iw) under different datasets. We present kernel density estimation (KDE) plots of the LOO estimator, the empirical miscoverage, and the empirical expectation miscoverage level. All results are obtained using a ResNet50 model under the specified size constraint rule with a calibration set size n = 200. Note: Distributions that are more co… view at source ↗

**Figure 2.** Figure 2: Comparison of LOO estimator convergence rate and stability between BCP and ST-BCP across different calibration set sizes n. We report the MSE and STD that are obtained using a ResNet50 model on the CIFAR-10 dataset under the size constraint rule T = 2. As n increases, our method ST-BCP (h = Iw) consistently outperforms the baseline BCP (h = s). mobilenetv3small efficientnetb0 resnet50 densenet121 0.00000 0… view at source ↗

**Figure 3.** Figure 3: Performance comparison between BCP(h = s) and ST-BCP(h = Iw) under different models. We report the MSE and GAP that are obtained on the CIFAR-10 dataset with the calibration set size of n = 200 under the size constraint rule T = 2. Among these models, our method ST-BCP(h = Iw) has always outperformed the baseline BCP(h = s). Evaluation metrics. We evaluate the performance of our method by measuring the fol… view at source ↗

**Figure 4.** Figure 4: Under different datasets, the MisCov, kernel density estimation plots of the empirical distribution of the LOO estimator, and the empirical miscoverage levels of both are presented. The results were obtained with the size constraint rule, using the ResNet50 model and the calibration set size n = 200. The validity of the original coverage guarantee is preserved as long as the covariance term remains non-neg… view at source ↗

**Figure 5.** Figure 5: Under different datasets, the MisCov, kernel density estimation plots of the empirical distribution of the LOO estimator, and the empirical miscoverage levels of both are presented. The results were obtained with the size constraint rule, using the ResNet50 model and the calibration set size n = 200. transformation to optimize the non-concentrated term h(wn+1; Dn+1, Xn+1) ; and (2) adopting a method that b… view at source ↗

read the original abstract

Conformal Prediction (CP) provides a statistical framework for uncertainty quantification that constructs prediction sets with coverage guarantees. While CP yields uncontrolled prediction set sizes, Backward Conformal Prediction (BCP) inverts this paradigm by enforcing a predefined upper bound on set size and estimating the resulting coverage guarantee. However, the looseness induced by Markov's inequality within the BCP framework causes a significant gap between the estimated coverage bound and the empirical coverage. In this work, we introduce ST-BCP, a novel method that introduces a data-dependent transformation of nonconformity scores to narrow the coverage gap. In particular, we develop a computable transformation and prove that it outperforms the baseline identity transformation. Extensive experiments demonstrate the effectiveness of our method, reducing the average coverage gap from 4.20\% to 1.12\% on common benchmarks.

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.

Desk Editor's Note private letter to a colleague

They introduce a data-dependent transformation of nonconformity scores that they prove tightens the BCP coverage bound over the identity map, with experiments showing a clear drop in the gap.

read the letter

The main takeaway is that this paper gives a computable, data-dependent transformation for nonconformity scores in backward conformal prediction. They prove it produces a strictly better lower bound on coverage than the plain identity transformation, and the experiments cut the average coverage gap from 4.20% to 1.12% on standard benchmarks. That combination of proof and concrete numbers is the part worth noting first. The transformation itself and the claim of superiority over identity look new relative to the BCP papers cited in the abstract. The work does a clean job of identifying the looseness that Markov's inequality introduces in BCP and then offering a fix that stays computable from the calibration set. The experiments appear to use common benchmarks and report the gap reduction directly, which makes the practical effect easy to see. The soft spot sits exactly where the stress-test note flags it. Because the transformation depends on the data, it risks changing the expectation or breaking the unconditional probability statement that Markov requires. If the transformed scores become dependent on the same points used to define the bound, the coverage guarantee could turn conditional or lose its validity. The abstract says they have a proof of outperformance, so they presumably adjust for this, but that derivation step is the one that needs the closest check. This is for people already working on conformal methods who care about controlling prediction set size in practice. A reader who wants a drop-in improvement to BCP would get usable value from the method and the reported numbers. I would send it to peer review. The empirical improvement is straightforward to evaluate and the central construction is specific enough that referees can test the validity argument directly.

Referee Report

2 major / 2 minor

Summary. The paper introduces ST-BCP, a method for Backward Conformal Prediction that applies a data-dependent transformation to nonconformity scores. The central claim is that this transformation is computable, provably yields a strictly tighter valid lower bound on coverage than the identity transformation via Markov's inequality, and reduces the empirical coverage gap from 4.20% to 1.12% on standard benchmarks.

Significance. If the validity of the coverage guarantee is preserved, the work would meaningfully improve the utility of BCP by narrowing the gap between the Markov-derived bound and actual coverage for fixed-size prediction sets. The reported experimental reduction is substantial and the claim of a computable transformation with a proof of improvement would be a clear strength if the derivation holds without hidden conditioning or bias.

major comments (2)

[Theoretical derivation of ST-BCP bound] The section deriving the coverage bound after introducing the transformation must explicitly verify that the data-dependent map leaves the relevant expectation unchanged (or correctly adjusted) so that Markov's inequality continues to apply unconditionally; the skeptic note correctly identifies this as the load-bearing step, and any conditioning on calibration statistics would turn the guarantee conditional and undermine the central claim.
[Proof of outperformance] The proof that the proposed transformation strictly outperforms the identity map (abstract and §3) should include the explicit inequality relating the two bounds; without it, the outperformance claim reduces to an empirical observation rather than a guaranteed improvement.

minor comments (2)

[Method description] Clarify the exact functional form of the transformation and whether it is computed solely from the calibration set or also involves test-point information; this affects both reproducibility and the exchangeability argument.
[Experiments] The experimental section should report the number of random seeds, exact benchmark datasets, and whether the coverage gap is measured as absolute or relative difference to allow direct replication of the 4.20% to 1.12% reduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The two major comments identify important points for strengthening the theoretical presentation. We address each below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: [Theoretical derivation of ST-BCP bound] The section deriving the coverage bound after introducing the transformation must explicitly verify that the data-dependent map leaves the relevant expectation unchanged (or correctly adjusted) so that Markov's inequality continues to apply unconditionally; the skeptic note correctly identifies this as the load-bearing step, and any conditioning on calibration statistics would turn the guarantee conditional and undermine the central claim.

Authors: We agree that an explicit verification is necessary for clarity. In the derivation, the transformation is a fixed (data-dependent) function of the calibration set alone. Because the test point is independent of the calibration set, the expectation of the transformed nonconformity score is taken unconditionally over the test-point distribution. We will revise Section 3 to insert a dedicated paragraph that states: let T be the transformation computed from the calibration set C; then E[T(S(X_{n+1},Y_{n+1}))] = E[T(S(X_{n+1},Y_{n+1})) | C] almost surely, so Markov’s inequality applies directly to the unconditional expectation. This removes any ambiguity about hidden conditioning. revision: yes
Referee: [Proof of outperformance] The proof that the proposed transformation strictly outperforms the identity map (abstract and §3) should include the explicit inequality relating the two bounds; without it, the outperformance claim reduces to an empirical observation rather than a guaranteed improvement.

Authors: We will make the comparison explicit. Let B_id denote the Markov bound obtained with the identity map and B_ST the bound obtained after the proposed transformation. The proof in §3 already establishes that the transformed scores satisfy a strictly smaller tail probability under the same Markov application whenever the transformation is non-constant on the support of the score distribution. We will add the direct inequality B_ST < B_id (with equality only in the degenerate case) immediately after the statement of the main theorem, together with the short algebraic step that shows the transformed expectation is smaller while the Markov multiplier remains identical. revision: yes

Circularity Check

0 steps flagged

Minor self-citation present but central derivation remains independent with explicit proof

full rationale

The paper develops a computable data-dependent transformation of nonconformity scores and supplies a proof that it strictly outperforms the identity map while preserving the Markov-based coverage bound. No equation reduces the claimed tighter bound to a fitted parameter or to the input data by construction. The transformation is presented as a new construction whose improvement is proven rather than assumed via self-citation. Any concern about exchangeability under data dependence is a validity/correctness question, not a circularity reduction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard conformal prediction assumptions such as data exchangeability and the use of Markov's inequality for the coverage bound; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption Data points are exchangeable
Standard assumption required for conformal prediction coverage guarantees, invoked implicitly for both BCP and the new transformation.
standard math Markov's inequality provides a valid (if loose) upper bound on coverage
The looseness addressed by the paper originates from this inequality applied to the nonconformity scores.

pith-pipeline@v0.9.0 · 5673 in / 1280 out tokens · 28581 ms · 2026-05-21T14:10:15.702904+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce ST-BCP, a novel method that introduces a data-dependent transformation of nonconformity scores to narrow the coverage gap... derive the optimal transformation under a monotonicity constraint... G(h)(s;D,X)=h(w(D,X);D,X)I(s≥w(D,X))
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the coverage bound in Eq. (7) is derived via Markov’s inequality, which only utilizes the expectation... ST-BCP reshapes the score distribution into a two-point structure

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 4 internal anchors

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Alami, N., Zakharia, J., and Taieb, S. B. Symmetric aggre- gation of conformity scores for efficient uncertainty sets. arXiv preprint arXiv:2512.06945,

work page arXiv

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Angelopoulos, A., Bates, S., Malik, J., and Jordan, M. I. Uncertainty sets for image classifiers using conformal prediction.arXiv preprint arXiv:2009.14193,

work page arXiv 2009

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Theoretical Foundations of Conformal Prediction

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work page internal anchor Pith review Pith/arXiv arXiv

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Balinsky, A. A. and Balinsky, A. D. Enhancing confor- mal prediction using e-test statistics.arXiv preprint arXiv:2403.19082,

work page arXiv

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End to End Learning for Self-Driving Cars

Bojarski, M., Del Testa, D., Dworakowski, D., Firner, B., Flepp, B., Goyal, P., Jackel, L. D., Monfort, M., Muller, U., Zhang, J., et al. End to end learning for self-driving cars.arXiv preprint arXiv:1604.07316,

work page internal anchor Pith review Pith/arXiv arXiv

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Csillag, D., Struchiner, C. J., and Goedert, G. T. Prediction- powered e-values.arXiv preprint arXiv:2502.04294,

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Conformal prediction for deep classifier via label ranking

Huang, J., Xi, H., Zhang, L., Yao, H., Qiu, Y ., and Wei, H. Conformal prediction for deep classifier via label ranking. arXiv preprint arXiv:2310.06430,

work page arXiv

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Batch multivalid conformal prediction.arXiv preprint arXiv:2209.15145,

Jung, C., Noarov, G., Ramalingam, R., and Roth, A. Batch multivalid conformal prediction.arXiv preprint arXiv:2209.15145,

work page arXiv

[10] [10]

Confor- mal prediction with learned features.arXiv preprint arXiv:2404.17487,

Kiyani, S., Pappas, G., and Hassani, H. Confor- mal prediction with learned features.arXiv preprint arXiv:2404.17487,

work page arXiv

[11] [11]

Koning, N. W. Post-hoc α hypothesis testing and the post- hoc p-value.arXiv preprint arXiv:2312.08040,

work page arXiv

[12] [12]

Conformal prediction with large language models for multi-choice question answering

9 ST-BCP: Tightening Coverage Bound for Backward Conformal Prediction via Non-Conformity Score Transformation Kumar, B., Lu, C., Gupta, G., Palepu, A., Bellamy, D., Raskar, R., and Beam, A. Conformal prediction with large language models for multi-choice question answering. arXiv preprint arXiv:2305.18404,

work page arXiv

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T., Doucet, A., et al

Stutz, D., Cemgil, A. T., Doucet, A., et al. Learning optimal conformal classifiers.arXiv preprint arXiv:2110.09192,

work page arXiv

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Api is enough: Conformal prediction for large language models without logit-access.arXiv preprint arXiv:2403.01216,

Su, J., Luo, J., Wang, H., and Cheng, L. Api is enough: Conformal prediction for large language models without logit-access.arXiv preprint arXiv:2403.01216,

work page arXiv

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Selective Conformal Risk Control

Xu, Y ., Guo, W., and Wei, Z. Selective conformal risk control.arXiv preprint arXiv:2512.12844,

work page internal anchor Pith review Pith/arXiv arXiv

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Image data augmentation for deep learning: A survey

Yang, S., Xiao, W., Zhang, M., Guo, S., Zhao, J., and Shen, F. Image data augmentation for deep learning: A survey. arXiv preprint arXiv:2204.08610,

work page arXiv

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mixup: Beyond Empirical Risk Minimization

Zhang, H., Cisse, M., Dauphin, Y . N., and Lopez-Paz, D. mixup: Beyond empirical risk minimization.arXiv preprint arXiv:1710.09412,

work page internal anchor Pith review Pith/arXiv arXiv

[18] [18]

When T is allowed to depend only on the test point feature X, the entropy is computed directly from the model’s softmax outputπ(X). Specifically, we define ENmax =log(|Y|), EN min = 0, EN(X) =− X y∈Y πy(X)log(π y(X)) Since this setting does not involve any dataset-level input D, the computation is a simplified special case of T(D, X) . Consequently, both ...

work page 2005