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arxiv: 2606.11136 · v2 · pith:46MCYOEK · submitted 2026-06-09 · math.ST · stat.ME· stat.ML· stat.TH

Conformal Prediction for Dyadic Regression Under Complex Missingness

Robert Lunde , Minjie Yang , Elizaveta Levina , Ji Zhu This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-27 11:04 UTCgrok-4.3pith:46MCYOEKrecord.jsonopen to challenge →

classification math.ST stat.MEstat.MLstat.TH
keywords conformal predictiondyadic regressionmissing datagraphon modelexchangeabilitynetwork datamissing not at random
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The pith

A weighted conformal procedure achieves asymptotic validity for dyadic regression under a nonparametric graphon model for missingness.

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

The paper develops conformal prediction methods for dyadic regression problems where data consist of observations on pairs and may exhibit complex missingness. It introduces general tools for proving finite-sample validity of conformal procedures under invariance conditions weaker than exchangeability, using a bijection argument to address cases in which the observed sample is itself a random subset of the index set. Several procedures are proposed for jointly exchangeable arrays, and a weighted conformal method is shown to deliver asymptotic validity for missing entries when the missingness mechanism follows a nonparametric graphon model, together with conditional validity guarantees for both continuous and discrete responses.

Core claim

The authors establish asymptotic validity of a weighted conformal prediction procedure for missing elements in dyadic data under a nonparametric graphon model for the missingness mechanism. They also provide the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption, for both continuous and discrete responses.

What carries the argument

A novel bijection argument constructing an explicit measure-preserving correspondence between events to handle random subsets of the index set, together with the weighted conformal procedure whose weights are derived from the graphon model.

If this is right

  • The full, split, row-column, and selective conformal procedures achieve finite-sample validity under the stated invariance conditions for jointly exchangeable arrays.
  • Asymptotic coverage guarantees apply to the weighted procedure when used on missing elements.
  • Conditional validity holds for both continuous and discrete response variables.
  • The methods apply directly to network data examples, both synthetic and real.

Where Pith is reading between the lines

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

  • If the graphon model matches the true missingness process in a given network, these intervals can be used to quantify uncertainty without invoking a missing-at-random assumption.
  • The bijection technique may apply to other array-valued or matrix-valued sampling problems that involve random subsets.
  • Empirical tests of whether observed missingness patterns are consistent with a graphon structure would be a direct way to assess applicability.

Load-bearing premise

The missingness mechanism follows a nonparametric graphon model that permits derivation of the weights and the coverage guarantee.

What would settle it

An empirical check in which the missingness probabilities are generated from a non-graphon mechanism and the observed coverage rate of the weighted intervals falls materially below the nominal level would refute the asymptotic validity claim.

Figures

Figures reproduced from arXiv: 2606.11136 by Elizaveta Levina, Ji Zhu, Minjie Yang, Robert Lunde.

Figure 1
Figure 1. Figure 1: Top row: coverage of the weighted conformal prediction interval for [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
read the original abstract

We develop a framework for conformal prediction in dyadic regression problems under complex missingness mechanisms. At the theoretical level, we develop general technical tools for establishing finite-sample validity of conformal prediction under distributional invariance conditions weaker than exchangeability. A key result handles the case where the sample itself is a random subset of the index set, a setting not covered by existing theory, via a novel bijection argument that constructs an explicit measure-preserving correspondence between events. In addition, we propose conformal prediction procedures for jointly exchangeable arrays, including full conformal, split conformal, a row-column approach exploiting similarities within rows and columns, and a selective conformal procedure achieving mask-conditional validity. For missing elements, we establish asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism. We further establish conditional validity results for both continuous and discrete responses; to the best of our knowledge, this is the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption. The proposed methods are illustrated on synthetic and real network data.

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 develops conformal prediction methods for dyadic regression under complex missingness. It introduces general tools for finite-sample validity under distributional invariance weaker than exchangeability, including a novel bijection argument for cases where the sample is a random subset of the index set. Procedures are proposed for jointly exchangeable arrays (full, split, row-column, and selective conformal). For missing elements under MNAR, it establishes asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism and claims the first formal proof of asymptotic conditional validity for weighted conformal prediction under MNAR, illustrated on synthetic and real network data.

Significance. If the central results hold, the work provides new technical tools for conformal prediction beyond exchangeability and extends coverage guarantees to MNAR settings in dyadic data via graphon modeling. The bijection argument for random subsets and the conditional validity claim under MNAR are potentially significant contributions for network applications, though the nonparametric estimation requirements for the weights remain a key point of evaluation.

major comments (2)
  1. [Abstract / theorem on asymptotic validity under graphon model] The asymptotic validity result for the weighted conformal procedure (abstract and the relevant theorem establishing coverage under the graphon model): the claim relies on plug-in weights derived from the nonparametric graphon estimator, but the manuscript does not state regularity conditions (e.g., Hölder smoothness, lower bounds away from zero, or uniform convergence rates) that would ensure the estimation error is small enough for the asymptotic coverage to hold, particularly for the mask-conditional result.
  2. [Section establishing conditional validity results] The conditional validity claim for weighted conformal prediction under MNAR (abstract): while the bijection argument is used for the invariance, the transition from finite-sample to asymptotic conditional validity under the fully nonparametric graphon appears to require additional justification that the estimated weights preserve the necessary conditional independence or invariance properties at the required rate.
minor comments (2)
  1. [Introduction / notation section] Notation for the dyadic array and the missingness indicators could be clarified with an explicit definition early in the paper to aid readability.
  2. [Section on row-column conformal procedure] The description of the row-column approach would benefit from a small illustrative example or diagram showing how row and column similarities are exploited.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The two major comments correctly identify gaps in the regularity conditions and justification for the asymptotic results. We will revise the manuscript to address both points by adding the necessary assumptions and expanding the proofs.

read point-by-point responses

  1. Referee: [Abstract / theorem on asymptotic validity under graphon model] The asymptotic validity result for the weighted conformal procedure (abstract and the relevant theorem establishing coverage under the graphon model): the claim relies on plug-in weights derived from the nonparametric graphon estimator, but the manuscript does not state regularity conditions (e.g., Hölder smoothness, lower bounds away from zero, or uniform convergence rates) that would ensure the estimation error is small enough for the asymptotic coverage to hold, particularly for the mask-conditional result.

    Authors: We agree that explicit regularity conditions are required for the plug-in estimator to yield the claimed asymptotic coverage. In the revision we will state Hölder smoothness of the graphon, a uniform lower bound away from zero on the missingness probabilities, and a uniform convergence rate for the nonparametric estimator. These assumptions will be added to the theorem statement together with a short argument showing that the resulting weight error is o_p(1) uniformly, which is sufficient for the coverage result to hold (including the mask-conditional version). revision: yes

  2. Referee: [Section establishing conditional validity results] The conditional validity claim for weighted conformal prediction under MNAR (abstract): while the bijection argument is used for the invariance, the transition from finite-sample to asymptotic conditional validity under the fully nonparametric graphon appears to require additional justification that the estimated weights preserve the necessary conditional independence or invariance properties at the required rate.

    Authors: The referee correctly notes that the passage from finite-sample invariance (via the bijection) to asymptotic conditional validity needs further detail once weights are estimated. We will expand the proof in the relevant section to show that, under the regularity conditions added in response to the first comment, the difference between the oracle and plug-in weights vanishes at a rate that preserves the conditional coverage property asymptotically. This will be done by controlling the total variation distance between the two weighted distributions conditionally on the mask. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent technical tools and relies on external modeling assumptions.

full rationale

The paper develops novel technical tools including a bijection argument for finite-sample validity under weaker distributional invariance conditions than exchangeability, and proposes multiple conformal procedures (full, split, row-column, selective) for jointly exchangeable arrays. Asymptotic validity for the weighted procedure is established under the nonparametric graphon model for missingness as an explicit modeling assumption used to derive weights and coverage guarantees, rather than any fitted parameter or self-referential reduction. No steps in the provided abstract or described claims reduce by construction to inputs via self-definition, fitted predictions, or load-bearing self-citations; the central results appear self-contained with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on distributional invariance conditions weaker than exchangeability and on a nonparametric graphon model for the missingness mechanism; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption distributional invariance conditions weaker than exchangeability
    Invoked to establish finite-sample validity of conformal prediction; stated as the key theoretical relaxation in the abstract.
  • domain assumption nonparametric graphon model for the missingness mechanism
    Used to derive weights and prove asymptotic validity of the weighted conformal procedure for missing elements.

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