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arxiv: 2606.28730 · v1 · pith:LYS62BZQnew · submitted 2026-06-27 · 🧮 math.ST · stat.ML· stat.TH

Full Conformal Prediction under Stochastic Non-Conformity Measure

Thanawat Sornwanee This is my paper

Pith reviewed 2026-06-30 08:59 UTC · model grok-4.3

classification 🧮 math.ST stat.MLstat.TH
keywords conformal predictionstochastic non-conformity measurepermutation invarianceconditional independencemachine learningcoverage guarantee
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The pith

The standard relaxation to permutation invariance in distribution does not guarantee validity of full conformal prediction when the non-conformity measure is stochastic.

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

Full conformal prediction has long assumed deterministic non-conformity measures, yet modern machine-learning training introduces unavoidable randomness. A frequently proposed relaxation—that the measure need only be invariant in distribution under data permutations—fails to ensure the required coverage property. The paper demonstrates this insufficiency with a counterexample and replaces it with a stronger sufficient condition: the measure must also be conditionally independent of the permutation given the data. This pair of requirements is shown to hold in several practical stochastic training regimes.

Core claim

We show that the condition of almost sure permutation invariance is too restrictive for stochastic non-conformity measures arising from machine learning, and that the relaxation to permutation invariance in distribution is insufficient to guarantee the validity of full conformal prediction. We provide a correct sufficient condition consisting of conditional independence together with permutation invariance in distribution, which is satisfied in several stochastic machine learning settings.

What carries the argument

Conditional Independence & Permutation Invariance in Distribution: the joint requirement that the stochastic non-conformity measure remains distributionally unchanged under data permutations and is independent of those permutations conditional on the observed data.

If this is right

  • Full conformal prediction retains its coverage guarantee under any stochastic training process obeying the corrected condition.
  • The condition is satisfied by several common sources of randomness in machine-learning pipelines.
  • Validity proofs no longer require the non-conformity measure to be deterministic or almost surely invariant.

Where Pith is reading between the lines

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

  • Implementations that add controlled randomness to training may still produce valid conformal sets if the independence structure is preserved.
  • The result highlights that dependence between training randomness and data ordering, rather than randomness alone, is what breaks coverage.
  • Testing the condition on concrete randomized learners such as bagged models or networks with dropout would give practical guidance on when the guarantee holds.

Load-bearing premise

The randomness introduced by machine-learning training can be captured by making the non-conformity measure conditionally independent of data permutations.

What would settle it

An explicit counterexample consisting of a finite data set, a stochastic training procedure, and a non-conformity score that is permutation-invariant in distribution but not conditionally independent of the permutation, for which the resulting prediction sets undercover below the nominal level.

Figures

Figures reproduced from arXiv: 2606.28730 by Thanawat Sornwanee.

Figure 1
Figure 1. Figure 1: DT stands for deterministic permutation invariance, which is a standard assumption in full conformal prediction as out￾lined in the subsection 4.1. (Vovk et al., 2005) AS stands for almost sure permutation invariance as outlined in the subsection 4.2 I&D stands for independence and permutation invariance in distribution as outlined in the subsection 4.3. CI&D stands for conditional independence and permuta… view at source ↗
read the original abstract

The theory of full conformal prediction uses deterministic non-conformity measure, but modern usage of full conformal prediction often relies on machine learning training, making stochasticity inevitable. A simple sufficient condition of almost sure permutation invariance of the non-conformity measure can be too restrictive, so many have suggested the relaxation to permutation in distribution as a condition for full conformal prediction validity. We, however, show that this commonly known condition is actually insufficient. We then provide a correct sufficient condition: Conditional Independence & Permutation Invariance in Distribution, which encompasses several stochastic settings that may be used in machine learning.

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 argues that the commonly invoked sufficient condition of permutation invariance in distribution for a stochastic non-conformity measure is insufficient to guarantee marginal coverage in full conformal prediction. It supplies a counterexample and then proposes the stronger condition of conditional independence between the randomness and the data together with permutation invariance in distribution; this pair is claimed to be sufficient and to cover typical stochastic training procedures in machine learning.

Significance. If the counterexample is valid and the sufficiency proof is correct, the result clarifies a subtle but practically relevant gap in the theory of conformal prediction with randomized predictors. The new condition is stated in a form that can be checked for common stochastic training schemes, which is a concrete advance over the literature that had treated distributional invariance as adequate.

major comments (2)
  1. [§3] §3 (counterexample construction): the joint distribution of the score vector must be shown to remain permutation-invariant while the realized coverage fails; the construction must explicitly keep the auxiliary randomness R independent of the test point (as required by the setup) rather than coupling R to the augmented data set. Without this verification the claimed insufficiency does not follow.
  2. [Theorem 4.1] Theorem 4.1 (sufficiency): the proof that conditional independence of R plus distributional invariance implies exchangeability of the realized scores is load-bearing; the argument should be written out in full rather than sketched, because any hidden dependence between R and the test point would invalidate the exchangeability step.
minor comments (2)
  1. Notation for the stochastic non-conformity measure A_n should be introduced once and used consistently; the current alternation between A and A_n is distracting.
  2. The abstract states the insufficiency result without any indication of the counterexample; a one-sentence pointer to the construction would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments, which help strengthen the presentation of the counterexample and the sufficiency result. We address the two major comments point by point below and will revise the manuscript to incorporate the requested clarifications and expansions.

read point-by-point responses

  1. Referee: [§3] §3 (counterexample construction): the joint distribution of the score vector must be shown to remain permutation-invariant while the realized coverage fails; the construction must explicitly keep the auxiliary randomness R independent of the test point (as required by the setup) rather than coupling R to the augmented data set. Without this verification the claimed insufficiency does not follow.

    Authors: We agree that the counterexample requires explicit verification to be fully rigorous. In the revised manuscript we will add a dedicated paragraph (or subsection) that (i) confirms the auxiliary randomness R is generated independently of the test point, (ii) shows that the joint distribution of the realized score vector remains permutation-invariant, and (iii) demonstrates that marginal coverage nevertheless fails. This will make the insufficiency claim self-contained. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (sufficiency): the proof that conditional independence of R plus distributional invariance implies exchangeability of the realized scores is load-bearing; the argument should be written out in full rather than sketched, because any hidden dependence between R and the test point would invalidate the exchangeability step.

    Authors: We accept that the current sketch of the proof is insufficient for a load-bearing result. In the revision we will replace the sketch with a complete, self-contained proof of Theorem 4.1. The expanded argument will explicitly invoke the conditional independence of R from the data (including the test point) at each step, derive the exchangeability of the score vector, and thereby obtain the marginal coverage guarantee. We will also add a short remark ruling out hidden dependence between R and the test point under the stated assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained

full rationale

The paper claims to exhibit insufficiency of 'permutation invariance in distribution' via counterexample and then states an independent sufficient condition (conditional independence plus distributional invariance). No equations, definitions, or self-citations in the abstract reduce the validity result to a fit, renaming, or prior author work; the new condition is introduced as an external strengthening rather than being defined in terms of coverage. The derivation chain therefore rests on external probabilistic arguments rather than internal self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard axioms of probability theory used to define distributional invariance and conditional independence for the non-conformity measure; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of probability theory including definitions of random variables, conditional independence, and distributional equality under permutations
    Invoked to formalize the stochastic properties of the non-conformity measure and the proposed condition.

pith-pipeline@v0.9.1-grok · 5615 in / 1165 out tokens · 48619 ms · 2026-06-30T08:59:59.191049+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

27 extracted references · 6 canonical work pages · 2 internal anchors

  1. [1]

    The Annals of Statistics , volume=

    Conformal prediction beyond exchangeability , author=. The Annals of Statistics , volume=. 2023 , publisher=

    2023

  2. [2]

    Symposium on conformal and probabilistic prediction with applications , pages=

    Criteria of efficiency for conformal prediction , author=. Symposium on conformal and probabilistic prediction with applications , pages=. 2016 , organization=

    2016

  3. [3]

    Conformal and Probabilistic Prediction and Applications , pages=

    Conformal testing in a binary model situation , author=. Conformal and Probabilistic Prediction and Applications , pages=. 2021 , organization=

    2021

  4. [4]

    Asian conference on machine learning , pages=

    Conditional validity of inductive conformal predictors , author=. Asian conference on machine learning , pages=. 2012 , organization=

    2012

  5. [5]

    arXiv preprint arXiv:2007.03114 , year=

    Efficient conformal prediction via cascaded inference with expanded admission , author=. arXiv preprint arXiv:2007.03114 , year=

    work page arXiv 2007

  6. [6]

    Theoretical Foundations of Conformal Prediction

    Theoretical foundations of conformal prediction , author=. arXiv preprint arXiv:2411.11824 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    conformal and probabilistic prediction and applications , pages=

    Universally consistent conformal predictive distributions , author=. conformal and probabilistic prediction and applications , pages=. 2019 , organization=

    2019

  8. [8]

    Electronic Journal of Statistics , volume=

    Training-conditional coverage for distribution-free predictive inference , author=. Electronic Journal of Statistics , volume=. 2023 , publisher=

    2023

  9. [9]

    Optimized conformal selection: Powerful selective inference after con- formity score optimization.arXiv preprint arXiv:2411.17983,

    Optimized conformal selection: Powerful selective inference after conformity score optimization , author=. arXiv preprint arXiv:2411.17983 , year=

    work page arXiv

  10. [10]

    Diversifying conformal selections

    Diversifying conformal selections , author=. arXiv preprint arXiv:2506.16229 , year=

    work page arXiv

  11. [11]

    Full-conformal novelty detection

    Full-conformal novelty detection: A powerful and non-random approach , author=. arXiv preprint arXiv:2501.02703 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    arXiv preprint arXiv:2306.06342 , year=

    Distribution-free inference with hierarchical data , author=. arXiv preprint arXiv:2306.06342 , year=

    work page arXiv

  13. [13]

    Journal of the American Statistical Association , volume=

    Distribution-free predictive inference for regression , author=. Journal of the American Statistical Association , volume=. 2018 , publisher=

    2018

  14. [14]

    2022 , eprint=

    A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification , author=. 2022 , eprint=

    2022

  15. [15]

    European conference on machine learning , pages=

    Inductive confidence machines for regression , author=. European conference on machine learning , pages=. 2002 , organization=

    2002

  16. [16]

    Advances in neural information processing systems , volume=

    Conformalized quantile regression , author=. Advances in neural information processing systems , volume=

  17. [17]

    Journal of the American Statistical Association , volume=

    Least ambiguous set-valued classifiers with bounded error levels , author=. Journal of the American Statistical Association , volume=. 2019 , publisher=

    2019

  18. [18]

    2026 , url=

    Elliot L Epstein and John Winnicki and Thanawat Sornwanee and Rajat Vadiraj Dwaraknath , booktitle=. 2026 , url=

    2026

  19. [19]

    2005 , publisher=

    Testing statistical hypotheses , author=. 2005 , publisher=

    2005

  20. [20]

    The Twelfth International Conference on Learning Representations , year=

    Conformal Language Modeling , author=. The Twelfth International Conference on Learning Representations , year=

  21. [21]

    Findings of the association for computational linguistics: ACL 2023 , pages=

    Conformal nucleus sampling , author=. Findings of the association for computational linguistics: ACL 2023 , pages=

    2023

  22. [22]

    Findings of the Association for Computational Linguistics: EMNLP 2024 , pages=

    API Is Enough: Conformal Prediction for Large Language Models Without Logit-Access , author=. Findings of the Association for Computational Linguistics: EMNLP 2024 , pages=

    2024

  23. [23]

    IFIP International conference on artificial intelligence applications and innovations , pages=

    Efficiency comparison of unstable transductive and inductive conformal classifiers , author=. IFIP International conference on artificial intelligence applications and innovations , pages=. 2014 , organization=

    2014

  24. [24]

    The Thirteenth International Conference on Learning Representations , year=

    Leave-One-Out Stable Conformal Prediction , author=. The Thirteenth International Conference on Learning Representations , year=

  25. [25]

    The Thirteenth International Conference on Learning Representations , year=

    Approximating Full Conformal Prediction for Neural Network Regression with Gauss-Newton Influence , author=. The Thirteenth International Conference on Learning Representations , year=

  26. [26]

    2018 , eprint=

    conformalClassification: A Conformal Prediction R Package for Classification , author=. 2018 , eprint=

    2018

  27. [27]

    2005 , publisher=

    Algorithmic learning in a random world , author=. 2005 , publisher=

    2005