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How to rank imputation methods?

T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that an energy-score-based imputation score ranks the true imputation distribution highest among all imputations compatible with the observed data, without needing the complete data, under its missingness assumption.

desk verdict Good idea, broken theorem as stated: the energy-I-Score needs CIMAR_j for every averaged component, not just one, and the proof swaps Mj=0/Mj=1. read the letter →

arxiv 2507.11297 v1 pith:LBOGM6PJ submitted 2025-07-15 stat.ME stat.AP

Jeffrey Näf , Krystyna Grzesiak , Erwan Scornet This is my paper · ORCID
classification stat.MEstat.AP MSC 62H1262G05
keywords energyscoreproperscoringrulesI-Scoreimputationrankingmissingatrandompattern-mixturemodelsmultipleconditionalindependence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Choosing an imputation method usually means benchmark studies that artificially mask more data and compare imputed values to the true ones with RMSE; the paper argues this is generally invalid under MAR and can systematically misrank methods. Its aim is to rank imputations for a given incomplete data set without access to the complete data, by scoring whether each method reproduces the conditional distribution of a missing variable given the variables that are always observed. To that end it builds the energy-I-Score from the energy distance and proves it is a proper I-Score under pattern-mixture MAR (PMM-MAR) plus a new condition, CIMAR$_j$, which says that for the scored variable $j$, the distribution of $X_j$ given the always-observed variables is the same in observed and missing patterns. In simulations and a downstream 401(k) analysis the score's ordering tracks the ordering one would get with the full data, while the earlier DR-I-Score is shown to fail in the paper's leading MAR example.

What carries the argument

The central object is the variable-wise energy score $S^j_{\mathrm{NA}}(H,P)$, built from the energy score $\mathrm{es}(P,y)=\frac{1}{2}\mathbb{E}_{X,X'\sim P}\lVert X-X'\rVert-\mathbb{E}_{X\sim P}\lVert X-y\rVert$, a strictly proper scoring rule that favors the distribution actually generating the test points. The construction masks the observed values of $X_j$, imputes them $N$ times from the candidate distribution $H_{X_j\mid X_{O_j},M_j=1}$, and compares these draws against the real observed values from $P^*_{X_j\mid X_{O_j},M_j=0}$. Propriety is carried by Condition CIMAR$_j$, which equates these two conditional distributions, turning the missing-data ranking problem into a distributional prediction problem on always-observed variables. Estimation uses the empirical energy-score formula of Equation (3.3) after $N$ multiple-imputation runs, with the per-variable scores averaged over the variables that can be missing.

What would settle it

Take the dependent-uniform example of Appendix B.1, where $X_1$ and $X_2$ are dependent and missingness depends on $X_1$, so CIMAR$_2$ fails; compute the population energy-I-Score. If any imputation distribution other than $P^*$ achieves a strictly higher averaged score, the theorem as stated is falsified because the averaged score includes a variable whose CIMAR condition fails.

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Extended reading notes

Core claim

The central claim is Proposition 3.1: under PMM-MAR and Assumption 3.1, the energy-I-Score $S_{\mathrm{NA}}$ is a proper I-Score, meaning that the true imputation distribution $P^*$ receives the highest expected score among all imputations compatible with the observed data. For each partly observed variable $j$, the score takes observed units where $X_j$ is present, artificially masks $X_j$, imputes it $N$ times from the candidate method given the always-observed variables $X_{O_j}$, and compares those draws to the real observed values with the energy score; the per-variable scores are averaged over the variables that can be missing. The propriety argument runs through Condition CIMAR$_j$, which states that $P^*_{X_j \mid X_{O_j}, M_j=0} = P^*_{X_j \mid X_{O_j}}$, so a correct imputation in missing patterns is indistinguishable, for scoring purposes, from the distribution that produced the observed values. The paper also shows the earlier classifier-based DR-I-Score requires the stronger EMAR condition and fails in the paper's leading MAR example, whereas the new score is proper there.

Load-bearing premise

The proof needs Condition CIMAR$_j$ to hold for every variable $j$ that contributes to the averaged score, whereas Assumption 3.1 only guarantees one such variable; if any scored variable violates it, that component can favor a wrong imputation and break propriety.

Editorial extensions

If this is right

  • Practitioners can select imputation methods for their own incomplete data by distributional fidelity, without needing a held-out complete sample or an auxiliary classifier.
  • The score deliberately ranks down methods that only impute conditional means, since near-identical draws across the $N$ replications score poorly on the energy score's spread term.
  • It extends proper I-Scores to settings like the paper's Example 1, where PMM-MAR holds but EMAR fails and the DR-I-Score is not proper.
  • Estimation is practical with roughly 20 to 50 imputation draws, and for high-dimensional data the score can be computed on a subset of variables with the most missingness.
  • In the downstream 401(k) study, the score's ordering agrees with which imputation brings double-machine-learning estimates and standard errors closest to the complete-data analysis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of Proposition 3.1 as written averages $S^j_{\mathrm{NA}}$ over all variables in $S$, but Assumption 3.1 only guarantees CIMAR$_j$ for one variable; a correct theorem needs the condition for every scored variable, or the average can be pulled down by a variable whose component score is improper.
  • A natural extension is to replace the univariate energy score with a multivariate energy distance over blocks of variables; the paper's own counterexample shows the univariate score cannot detect whether the joint dependence between two missing variables is preserved, so a multivariate version would restore that sensitivity at the price of sample efficiency.
  • When no variable is always observed, the implementation's fallback rule (use the variable most often jointly observed) lies outside the propriety theorem; applying the score in such designs is best treated as heuristic unless the condition is checked.
  • A testable practical prediction: restricting the average to variables that satisfy CIMAR$_j$ should reproduce the full-data ordering more faithfully than the unrestricted average in data sets where the condition holds for some variables but not all.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper proposes a new score, the energy-I-Score, for ranking imputation methods when the complete data are unavailable. The score is constructed variable by variable: for each variable that can be missing, it compares the candidate imputation distribution conditional on the always-observed variables (when the variable is missing) with observed test points of that variable, using the energy score. The authors identify a missingness condition, CIMAR_j, and state in Proposition 3.1 that under this condition for at least one j and under PMM-MAR the averaged score over all eligible variables is a proper I-Score. They provide an estimation algorithm using multiple imputation, compare the score with the DR-I-Score, and report simulations plus a downstream double-machine-learning application.

Significance. If the main theoretical claim were correct, this would be a practically valuable contribution: a simple, classifier-free, distribution-oriented tool for ranking imputations, directly addressing the known shortcomings of RMSE-based benchmarking and of the classifier-based DR-I-Score. The paper is clearly written, the simulations are informative, and the code is made available. The energy score component is standard and the estimation procedure is intuitive. However, the central theoretical result, Proposition 3.1, is false as stated; the gap is load-bearing for the paper's main claim. The simulations and downstream experiment appear consistent with a corrected version of the theorem, so the issue is fixable, but the manuscript in its current form overstates what is proved.

major comments (1)
  1. [Section 3.1, Proposition 3.1; Appendix C] Proposition 3.1 is false as stated. Assumption 3.1 asserts only the existence of at least one j for which CIMAR_j holds, while S_NA averages over all j in S. The proof in Appendix C establishes, at most, propriety of the single component S_j^NA for a j satisfying CIMAR_j; the final averaging step then requires CIMAR_j for every j in S. In addition, the proof applies the energy-score propriety inequality to H_{Xj|XOj,Mj=0} in Eq. (C.1), whereas the score in Eq. (3.2) is defined with H_{Xj|XOj,Mj=1}. This is not a mere typo: for H in H_P, compatibility forces H_{Xj|XOj,Mj=0} = P*_{Xj|XOj,Mj=0}, so the inequality for the Mj=0 version is vacuous and says nothing about the imputation actually scored. The dependent uniform example in Section B.1 (with patterns m1=(0,0,0), m2=(0,1,0), m3=(1,0,0) and dependence between X1 and X2) satisfies PMM-MAR and Assumption 3.1 (since CIMAR_1 holds) while CIMAR_2 fails. For that example, an H that imputes X1 as P* does and imputes X2 in pattern m2 from P*_{X2|X3,M2=0} instead of P*_{X2|X3,M2=1} is compatible with the observed distribution, has S_1^NA(H)=S_1^NA(P*), and, by strict propriety of the energy score, has S_2^NA(H)>S_2^NA(P*), contradicting the proposition. The claim becomes true if Assumption 3.1 is replaced by the condition that CIMAR_j holds for every j in S, or if the set S is restricted to indices satisfying CIMAR_j; neither statement appears in the manuscript, and the practical scope of the method changes substantially under either fix.
minor comments (4)
  1. [Section 3.2, Eq. (3.3)] The text describing the generated samples contains a typographical error: it writes 'X~(l)_i,j, ..., X~(l)_N' where the last subscript should be 'i,j' rather than 'N'. This is a presentation issue but could confuse readers implementing the estimator.
  2. [Section 3.2 and Algorithm 1] The estimation procedure is described as approximating H_{Xj|XOj,Mj=1} by imputing a concatenated dataset that includes the candidate method's own previously imputed values from patterns in Lc_j. This is a reasonable heuristic, but the paper does not discuss how the approximation error depends on the imputation method's internal iterative structure. A brief discussion or reference to the multiple-imputation literature would help.
  3. [References] The reference list contains two entries for Stekhoven and Buhlmann (2011), one written 'Stekhoven, D. J. and Buhlmann, P.' and another 'Stekhoven, D. J. and B\"uhlmann, P.'; these should be consolidated into a single entry with consistent accent formatting.
  4. [Section 5.1 and Figure 2] The figure caption and text refer to 'runifsq' as drawing from p*(x1|x2,x3,M=m1)=2x1, but the formula in the text uses an indicator notation that is somewhat terse; adding an explicit density expression would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the energy-I-Score's propriety is derived from the external energy-score propriety plus explicitly stated assumptions; the main weakness is a proof gap (Assumption 3.1 vs. averaging over S), which is a correctness risk, not circularity.

full rationale

The derivation of Proposition 3.1 does not reduce to its inputs. The score S_NA is defined independently of P* in Section 3.1, Eq. (3.2), with an empirical counterpart in Eqs. (3.3)-(3.4). Propriety is obtained by applying the energy-score propriety of Gneiting and Raftery (2007) to each component S_j^NA, then using CIMAR_j to identify P*_{Xj|XOj,Mj=1} with P*_{Xj|XOj,Mj=0}; the assumption and the conclusion are separate mathematical objects, and no fitted parameter is renamed as a prediction. The paper does build on Näf et al. (2023, 2025), but the load-bearing pieces are restated or re-proven: Definition 2.1 restates the I-Score definition, Proposition 2.1 is proven in Appendix C using PMM-MAR directly, and the new score's propriety relies on the external energy-score theorem rather than on an imported uniqueness claim. No ansatz is smuggled in via citation: CIMAR_j is introduced in Definition 3.2 and its relationship to RMAR and PMM-MAR is discussed in the paper itself. The notable weakness in the manuscript is a proof gap rather than circularity: Assumption 3.1 only asserts existence of one index j satisfying CIMAR_j, while S_NA averages over all j in S, and the Appendix C proof's summing step requires the equality for every averaged component; the skeptic's counterexample would make Proposition 3.1 false as stated. That is a correctness/proof risk, not a reduction of the conclusion to the inputs. Overall, the derivation is self-contained against external benchmarks and merits score 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The score itself has no fitted parameters; the only user-specified quantity is the Monte Carlo size N. The load-bearing assumptions are the missingness conditions (PMM-MAR plus the new CIMAR_j) and the consistency of the self-training estimation step. No new physical or structural entities are posited.

free parameters (1)
  • N = 50
    Number of imputation samples per test point, chosen by hand for experiments. Appendix B.3 shows rankings are stable for N >= 20 and even N=5 works in the examples.
assumptions (5)
  • domain assumption PMM-MAR holds for the data generating process.
    Invoked in Proposition 3.1 and throughout the paper.
  • ad hoc to paper CIMAR_j holds for every variable j entered into the score.
    The proof of Proposition 3.1 silently requires this for all j in S, although Assumption 3.1 only states existence of one such j.
  • domain assumption Equal support X|m = X|m' for all patterns (Assumption 2.1).
    Used throughout to avoid extrapolation outside observed supports.
  • standard math Strict propriety of the energy score (Gneiting and Raftery 2007).
    The central inequality in Proposition 3.1 is inherited from this standard result.
  • domain assumption Repeatedly imputing the masked dataset with H produces samples from H_{Xj|XOj,Mj=1}.
    Used in Section 3.2 estimation; not proven, and could fail if H's imputation behavior changes when trained on data containing its own imputed values.

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Pith. "Pith review of How to rank imputation methods?." pith.science (2026). https://pith.science/paper/LBOGM6PJ

@misc{pith2026250711297,
  author       = {Pith},
  title        = {Pith review of: How to rank imputation methods?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBOGM6PJ}},
  note         = {Machine review of arXiv:2507.11297}
}
read the original abstract

Imputation is an attractive tool for dealing with the widespread issue of missing values. Consequently, studying and developing imputation methods has been an active field of research over the last decade. Faced with an imputation task and a large number of methods, how does one find the most suitable imputation? Although model selection in different contexts, such as prediction, has been well studied, this question appears not to have received much attention. In this paper, we follow the concept of Imputation Scores (I-Scores) and develop a new, reliable, and easy-to-implement score to rank missing value imputations for a given data set without access to the complete data. In practice, this is usually done by artificially masking observations to compare imputed to observed values using measures such as the Root Mean Squared Error (RMSE). We discuss how this approach of additionally masking observations can be misleading if not done carefully and that it is generally not valid under MAR. We then identify a new missingness assumption and develop a score that combines a sensible masking of observations with proper scoring rules. As such the ranking is geared towards the imputation that best replicates the distribution of the data, allowing to find imputations that are suitable for a range of downstream tasks. We show the propriety of the score and discuss an estimation algorithm involving energy scores. Finally, we show the efficacy of the new score in simulated data examples, as well as a downstream task.

Figures

Figures reproduced from arXiv: 2507.11297 by the authors.

Figure 1
Figure 1. Conceptual illustration of the score approximation. First, the imputed values in blue are used to learn HXj |XOj ,Mj=1. Then, for each xi,Oj for which xi,j is observed, we score the “prediction” HXj |XOj ,Mj=1 using the energy score with test point xi,j . In practice, this is done by (approximately) generating a sample X˜ (l) i,j , l = 1, . . . , N from HXj |xi,Oj ,Mj=1. divergence DKL(HX|M=m || P ∗ X|M=0). Specific… view at source ↗
Figure 2
Figure 2. (A) Illustration of Example 1. Left: Distribution we would like to impute X ∗ 1 | M = m3. Middle: Distribution of X1 in the fully observed pattern (X1 | M = m1). Right: Distribution of all patterns for which X1 is observed (Mixture of the distribution of X1 in patterns m1 and m2). (B) Standardized scores for different imputations methods. Methods are ordered according to the mean score. 0 25 50 75 100 −10 0 10 X1 0 … view at source ↗
Figure 3
Figure 3. Results for the Gaussian mixture model with distribution shift. (A) Illustration of (X1, X2, X3); (B) Standardized scores for different imputations methods. Methods are ordered ac￾cording to the mean score. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Results for the Gaussian nonlinear mixture model. (A) Illustration of (X1, X2, X3), (B) Standardized scores for the Gaussian nonlinear mixture model. Methods are ordered according to the mean score. 6 Empirical Study: Downstream Tasks To further illustrate the use of t…
Figure 5
Figure 5. Figure 5: (A) Energy-I-Score averaged across all replications. (B) Result of the DML approach based on different machine learning methods and different imputation methods, over 10 imputations, for the point estimation ˆθ0 (above) and standard error estimation SE( ˆθ0) below. In …
Figure 6
Figure 6. Figure 6: Standardized scores for different imputations methods for the uniform example with depen￾dence. Methods are ordered according to the mean score. k = 1, . . . , N, represents one draw from the conditional distribution of the one-hot encoded representation of Xj given XO…
Figure 7
Figure 7. Figure 7: Illustration of the imputation using Gaussian independent distribution (rnorm) and Gaus￾sian conditional distribution (using mice norm.nob) in the Strict Propriety Counter Example. m1 = (1, 0, 0, 0, 0, 0), m2 = (0, 1, 0, 0, 0, 0), m3 = (0, 0, 0, 0, 0, 0). Each observat…
Figure 8
Figure 8. Figure 8: Standardized scores for the Strict Propriety Counter Example. Methods are ordered accord￾ing to the mean score. -0.250 -0.225 -0.200 -0.175 1 5 10 20 30 40 50 60 70 80 90 100 N energy-I-Score method mice CART mice DRF mice norm.nob mice norm.predict mice rf random runi…
Figure 9
Figure 9. Figure 9: Results from the uniform example in Section 5.1 for different N over 10 repetitions. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Results from the Gaussian mixture example in Section 5.3 for different N over 10 repeti￾tions. Proof. N¨af et al. (2025) showed that for all x ∈ X such that p ∗ (x−j | Mj = 0) > 0, p ∗ (xj | x−j , Mj = 0) = p ∗ (xj | x−j ). Moreover, it follows from PMM-MAR directly t…
Figure 11
Figure 11. Figure 11: Conceptual illustration of the score approximation for a random draw of a pattern M ∈ Lj , similarly to [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: (1) Gaussian Mixture Example of Section 5.2, (2) Nonlinear Mixture Example of Section 5.3, (3) Independent Uniform Example of Section 5.1, (4) Non-Strict Propriety Example of Section B.2 and (5) Dependent Uniform Example of Section B.1 with the new score. 31 [PITH_FU…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.