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arxiv: 2606.24959 · v1 · pith:2DKKZI22new · submitted 2026-06-23 · 💻 cs.LG · cs.AI

Reliable Conformal Prediction for Ordinal Classification Using the Ranked Probability Score

Stefan Haas , Luca Killmaier , Alireza Javanmardi , Eyke H\"ullermeier This is my paper

Pith reviewed 2026-06-26 00:41 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords conformal predictionordinal classificationranked probability scoreprediction setsnonconformity measureuncertainty quantificationproper scoring rules
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The pith

The ranked probability score as nonconformity measure produces median-centered contiguous prediction sets by construction in conformal prediction for ordinal classification.

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

The paper develops a conformal prediction method for ordinal classification that treats the ranked probability score as the nonconformity function. This choice generates prediction sets that remain contiguous and centered on the median without any post-processing steps. The approach supplies distribution-free marginal coverage while reflecting the ordered severity of errors, which matters in domains where misclassifications carry different costs. Experiments on image and tabular datasets indicate that the resulting sets achieve a practical balance between their width and the size of ordinal miscoverage relative to prior conformal methods for ordered outcomes.

Core claim

When the ranked probability score serves as the nonconformity measure, conformal prediction for ordinal classification yields median-centered contiguous prediction sets by construction. The procedure remains model-agnostic, applies to assessed or grouped ordered categories, and runs efficiently without greedy interval search, while experiments show competitive trade-offs between set size and ordinal error magnitude.

What carries the argument

The ranked probability score used as a nonconformity measure, which by construction produces contiguous and median-centered prediction sets.

If this is right

  • Prediction sets remain contiguous without any additional selection procedure.
  • Sets are centered on the median prediction by design.
  • The method applies directly to both assessed and grouped ordered outcomes.
  • Implementation is more efficient than greedy interval selection methods.
  • Sets achieve a balance between width and the magnitude of ordinal miscoverage.

Where Pith is reading between the lines

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

  • The construction could simplify deployment of ordinal conformal methods in practice by removing the need for custom contiguity enforcement.
  • Similar scoring rules that respect cumulative distributions might be tested for analogous automatic properties in other structured prediction settings.
  • High-stakes applications could measure whether the median-centering reduces costly ordinal errors more than width-minimizing alternatives.

Load-bearing premise

The ranked probability score functions as an effective nonconformity measure that automatically enforces contiguity and median-centering without further assumptions or post-processing.

What would settle it

Finding a dataset and model where RPS-based conformal sets are neither contiguous nor median-centered would falsify the construction claim.

Figures

Figures reproduced from arXiv: 2606.24959 by Alireza Javanmardi, Eyke H\"ullermeier, Luca Killmaier, Stefan Haas.

Figure 1
Figure 1. Figure 1: (Left) Illustration of an assessed ordered categorical variable (survival prognosis for melanoma [National Cancer [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of prediction sets at α = {0.01, 0.02, 0.03, 0.05, 0.08, 0.1, 0.13, 0.15, 0.18, 0.2} across methods and datasets using the MAMM metric. Shaded regions indicate standard deviation over 50 trials. 0.0 2.0 4.0 WAMM FGNet 1.0 2.0 3.0 4.0 RetinaMNIST 0.0 1.0 2.0 3.0 BACH 1.0 2.0 3.0 4.0 Wine Quality (Red) 0.05 0.10 0.15 0.20 α 0.0 2.0 4.0 WAMM ERA 0.05 0.10 0.15 0.20 α 0.5 1.0 1.5 2.0 SWD 0.05 0.10 0… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of prediction sets at α = {0.01, 0.02, 0.03, 0.05, 0.08, 0.1, 0.13, 0.15, 0.18, 0.2} across methods and datasets using the WAMM metric. Shaded regions indicate standard deviation over 50 trials [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of prediction sets at α = {0.01, 0.02, 0.03, 0.05, 0.08, 0.1, 0.13, 0.15, 0.18, 0.2} across methods and datasets using the AISL metric. Shaded regions indicate standard deviation over 50 trials. (BACH [Aresta et al., 2019] and RetinaMNIST [Yang et al., 2023]) and an age-estimation dataset (FGNet [Lanitis et al., 2002, Panis et al., 2016]). Additionally, we include multiple ordinal tabular benchm… view at source ↗
Figure 5
Figure 5. Figure 5: Example images from the BACH dataset [Aresta et al., 2019]. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example images from the RetinaMNIST dataset [Yang et al., 2023] [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example images from the FGNet dataset [Lanitis et al., 2002] [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of prediction sets at α = {0.01, 0.02, 0.03, 0.05, 0.08, 0.1, 0.13, 0.15, 0.18, 0.2} across methods on the BACH dataset [Aresta et al., 2019]. Shaded regions indicate standard deviation over 50 trials. 0.75 0.80 0.85 0.90 0.95 1.00 COV 0.0 0.1 0.2 0.3 CV 2 3 4 5 PS 1 2 3 4 MW 0.05 0.10 0.15 0.20 α 1.00 1.25 1.50 1.75 2.00 MAMM 0.05 0.10 0.15 0.20 α 1 2 3 4 WAMM 0.05 0.10 0.15 0.20 α 4 5 6 7 AISL… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of prediction sets across methods on the RetinaMNIST dataset [Yang et al., 2023]. Shaded regions [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of prediction sets across methods on the FGNet dataset [Lanitis et al., 2002]. Shaded regions indicate [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of prediction sets across methods on LESTSensors dataset. Shaded regions indicate standard [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of prediction sets across methods on LEVXSensors dataset. Shaded regions indicate standard [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of prediction sets across methods on nhanes dataset. Shaded regions indicate standard deviation. [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of prediction sets across methods on swd dataset. Shaded regions indicate standard deviation. [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of prediction sets across methods on winequalityRed dataset. Shaded regions indicate standard [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of prediction sets across methods on insurance dataset. Shaded regions indicate standard deviation. [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of prediction sets across methods on melbourneAirbnb dataset. Shaded regions indicate standard [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of prediction sets across methods on cancerDeathRate dataset. Shaded regions indicate standard [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of prediction sets across methods on era dataset. Shaded regions indicate standard deviation. [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison of prediction sets across methods on lev dataset. Shaded regions indicate standard deviation. [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Mean predictive probabilities (MP) for the different datasets with the CE loss. [PITH_FULL_IMAGE:figures/full_fig_p028_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Comparison of prediction sets across methods on heartDisease dataset using LightGBM. Shaded regions indicate [PITH_FULL_IMAGE:figures/full_fig_p030_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of prediction sets across methods on mammoexp dataset using LightGBM. Shaded regions indicate [PITH_FULL_IMAGE:figures/full_fig_p030_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Comparison of prediction sets across methods on support dataset using LightGBM. Shaded regions indicate [PITH_FULL_IMAGE:figures/full_fig_p030_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Comparison of prediction sets across methods on winequalityRed dataset using LightGBM. Shaded regions [PITH_FULL_IMAGE:figures/full_fig_p031_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Comparison of prediction sets across methods on nhanes dataset using LightGBM. Shaded regions indicate [PITH_FULL_IMAGE:figures/full_fig_p031_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Comparison of prediction sets across methods on LEVXSensors dataset using LightGBM. Shaded regions [PITH_FULL_IMAGE:figures/full_fig_p031_27.png] view at source ↗
read the original abstract

Ordinal classification (OC) arises in high-stakes domains such as medicine and finance, where uncertainty quantification must account for the severity of ordinal errors. Conformal prediction (CP) provides distribution-free prediction sets with marginal coverage guarantees; however, its practical effectiveness depends critically on the choice of nonconformity function. We introduce a CP method for ordinal classification based on the ranked probability score (RPS), a proper scoring rule defined over cumulative predictive distributions. Although it reflects ordinal risk quite naturally, it has largely been neglected in conformal ordinal prediction (COP). When used as a measure of nonconformity, RPS yields median-centered contiguous prediction sets by construction. The method is model-agnostic, supports both assessed and grouped ordered categorical outcomes, and permits efficient implementation compared to greedy interval selection procedures. Across multiple ordinal image and tabular datasets, RPS-based CP produces contiguous prediction sets and strikes a favorable balance between prediction set width and the magnitude of ordinal miscoverage relative to existing CP methods.

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

0 major / 2 minor

Summary. The manuscript proposes using the ranked probability score (RPS) as a nonconformity measure within conformal prediction for ordinal classification. It claims that this produces median-centered contiguous prediction sets by construction under standard exchangeability, without post-processing or extra assumptions. The method is model-agnostic, supports assessed and grouped outcomes, and is claimed to be computationally efficient. Empirical evaluations on ordinal image and tabular datasets are said to show a favorable balance between prediction-set width and ordinal miscoverage relative to prior conformal methods.

Significance. If the construction holds, the result supplies a parameter-free, proper-scoring-rule-based nonconformity measure that automatically enforces contiguity and median centering while retaining the usual marginal coverage guarantee. This is a practical advantage for ordinal problems in medicine and finance. Credit is due for the explicit verification that RPS sublevel sets are intervals containing the median (shown via contradiction for K=3 and K=4) and for the absence of invented entities or circular parameters.

minor comments (2)
  1. The abstract states that RPS 'yields median-centered contiguous prediction sets by construction,' yet the provided verification is limited to K=3 and K=4; a compact general argument (or reference to the known monotonicity of RPS away from the median) would make the claim self-contained for arbitrary K.
  2. The efficiency claim relative to 'greedy interval selection procedures' is asserted but not quantified; adding a brief complexity comparison or runtime table would strengthen the practical contribution without altering the central result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The feedback highlights the practical advantages of the RPS nonconformity measure, which aligns with our claims.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central result—that RPS as nonconformity score produces median-centered contiguous sets by construction—follows from the explicit mathematical property of the ranked probability score (minimized at the median of any fixed CDF F, with sublevel sets forming intervals), which is verified directly for small K without reference to fitted parameters from the target data or self-citations. Coverage remains the standard marginal guarantee under exchangeability; no step reduces a claimed prediction to an input by definition or via load-bearing self-citation. The empirical comparisons are separate and do not affect the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method inherits standard conformal prediction assumptions (exchangeability of calibration and test points) and treats RPS as a proper scoring rule whose properties are taken from prior literature; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Data points are exchangeable, enabling marginal coverage guarantees in conformal prediction.
    Standard assumption invoked for all conformal methods; required for the coverage claim.
  • standard math Ranked probability score is a proper scoring rule that penalizes ordinal distance appropriately.
    Taken from scoring-rule literature; used to justify the nonconformity choice.

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