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REVIEW 2 major objections 5 minor 34 references

Score Matching With Missing Data

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper adapts score matching to training data with coordinates missing completely at random, proving that minimising a marginal score-matching objective recovers the true full-data score, and giving two practical algorithms.

desk verdict Solid IW-based score matching for MCAR data, but the variational proof has a genuine θ-gradient error and the 'any score model' claim is overstated. read the letter →

arxiv 2506.00557 v1 pith:GFAP6GWT submitted 2025-05-31 stat.ML cs.LGstat.ME

Josh Givens , Song Liu , Henry W J Reeve This is my paper
classification stat.MLcs.LGstat.ME
keywords scorematchingmissingdatacompletelyatrandomimportanceweightingvariationalinferenceFisherdivergenceGaussiangraphicalmodelsunnormalizeddensity
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

Score matching normally needs fully observed data: it learns the gradient of a log-density by comparing model scores with data scores. This paper asks what happens when every training sample is missing some coordinates, under the missing-completely-at-random assumption. It establishes that minimising a marginal score-matching objective—one that compares scores only on the observed coordinates—recovers the true full-data score, provided the score model is the gradient of a known unnormalised density and marginal agreement implies joint agreement. Building on this, it gives an importance-weighting algorithm with finite-sample guarantees on bounded domains, and a variational algorithm that scales to higher-dimensional and graph-structured problems. The upshot is a way to train explicit parametric score models, not just diffusion networks, from partially observed data.

What carries the argument

The central object is the marginal score function $s_\lambda(x_\lambda) = \nabla_{x_\lambda}\log \int q(x)\,dx_{-\lambda}$ for an unnormalised density $q$, with $s_{\lambda;\theta}$ defined analogously. Proposition 4.3 shows the marginal Fisher divergence $F_M(\theta)=E[\|s_\Lambda(X_\Lambda)-s_{\Lambda;\theta}(X_\Lambda)\|^2]$ equals the computable objective $L_M(\theta)$ plus a constant. The importance-weighting estimator replaces the intractable marginal score integral by a self-normalised Monte Carlo ratio. Lemma 4.8 supplies the gradient-first identities $s_{\theta;\lambda}(x_\lambda)=E'[s_\theta(x_\lambda,X'_{-\lambda})_\lambda]$ and $\nabla E'[h_\theta]=E'[\nabla h_\theta]+\mathrm{Cov}'(s_\theta,h_\theta)$, which turn the objective into expectations and covariances under the model's conditional $p_\theta(\cdot|x_\lambda)$ and permit black-box variational approximation.

What would settle it

Take two different two-dimensional densities with identical one-dimensional marginals, for example a Gaussian and a mixture with the same marginals but different dependence. If missingness only ever leaves one coordinate observed, the marginal objective cannot distinguish them; checking whether the minimiser is the true joint density would directly test Proposition 4.3, and the recovery should fail whenever the identifiability assumption is violated.

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Extended reading notes

Core claim

The paper claims that for missing-completely-at-random data, the marginal Fisher divergence between the model's marginal score and the true marginal score is equivalent—up to a constant—to an expectation that depends only on the model, so it can be minimised from partial observations. Under an identifiability assumption, any minimiser of this marginal objective is the true unnormalised density almost surely. For truncated domains, the importance-weighted version of the objective has a high-probability finite-sample bound: the marginal truncated Fisher divergence of the estimated score decays at roughly $\sqrt{\log(dnrA/\delta)/\min\{r,n\}}$. The variational variant re-expresses the gradient of the objective in terms of conditional expectations and covariances under the model, which can be approximated by a learned conditional distribution, and is the variant that performs best in high-dimensional graphical model experiments.

Load-bearing premise

The whole construction assumes your score model is the gradient of a known unnormalised density $q_\theta$, so the marginal score can be written as an integral of that density; if you only have a black-box score network with no anti-derivative, neither algorithm can be run.

Editorial extensions

If this is right

  • Explicit parametric score models—Gaussian scores, energy-based models, and graphical-model precisions—can be trained on missing-completely-at-random data without imputation, by plugging $q_\theta$ into either algorithm.
  • With the importance-weighting method on bounded domains, choosing $r$ proportional to $n$ importance samples per observation gives roughly $1/\sqrt{n}$ error in the marginal truncated Fisher divergence, at an $O(n^2)$ computational cost.
  • Both algorithms extend to truncated, sliced, and denoising score matching, so the same marginalisation machinery can reach diffusion-style objectives.
  • In the paper's experiments, the variational variant recovers Gaussian graphical model edges from missing data with higher AUC than zero-imputation baselines, especially for structured graphs and higher dimensions.

Reading between the lines

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

  • If the missingness patterns only expose marginals that do not pin down the joint distribution, the identifiability assumption fails; a practical diagnostic would be to check whether the observed-coordinate marginals determine the model family before trusting the recovered full-data score.
  • The finite-sample guarantee is for the marginal objective, not the full-data score; a natural testable extension is to measure full-score error directly and see whether the same convergence rate survives.
  • The requirement of a known unnormalised density $q_\theta$ suggests a next step: parameterising neural scores as gradients of a potential network, or learning a separate potential, would let the same objectives apply to general black-box score models.
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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

2 major / 5 minor

Summary. The paper extends score matching to missing data under the missing-completely-at-random assumption. It defines marginal scores and shows in Proposition 4.3 that the marginal Fisher divergence can be optimized through an integration-by-parts objective that involves only the marginal score model and its divergence. Since marginal scores are intractable, the authors propose an importance-weighting estimator of the marginal score (Section 4.2, Algorithm 1) and a variational estimator that approximates the conditional expectations and covariances in the parameter gradient (Section 4.3, Algorithm 2, Corollary 4.9). The paper further derives finite-sample high-probability bounds for truncated importance-weighted score matching (Theorem 4.6 and Appendix A.15), extends the framework to sliced, denoised, and missing-not-at-random settings, and reports experiments on truncated Gaussian, non-Gaussian, Gaussian graphical model, S&P 100, and yeast data.

Significance. The population-level equivalence in Proposition 4.3 is a clean and useful extension of the classical score-matching identity to marginal observations, and the finite-sample bound in Theorem 4.6/Appendix A.15 is more detailed than is typical for such estimators. The paper ships code and is candid about an important limitation: the bounds are for the marginal Fisher divergence, not for the full-data score error (Remark 4.7 and Section 6). However, the advertised scope of any parameterised score model exceeds what the algorithms actually require, and the variational method rests on Lemma 4.8, which is false as stated for θ-gradients. Both issues are fixable, but they need to be addressed before the claims can be accepted.

major comments (2)
  1. [Lemma 4.8; Appendix C.3; Corollary 4.9; Lemma A.18] The identity in Lemma 4.8, Eq. (9), is stated for ∇ equal to either ∇_{x_λ} or ∇_θ, but it is false for ∇ = ∇_θ. For ∇ = ∇_θ, the log-derivative computation gives ∇_θ E'[h_θ] = E'[∇_θ h_θ] + Cov'(∇_θ log q_θ, h_θ), not Cov'(s_θ, h_θ) with s_θ = ∇_x log q_θ. In the proof in Appendix C.3, the step replacing s_{θ;λ}(x_λ) E'[g_θ] by E'[∇ log p_θ(x_λ, X'_−λ)] E'[g_θ] is valid only when ∇ is the x_λ-gradient; for ∇ = ∇_θ the correct factor is E'[∇_θ log p_θ] E'[g_θ]. The same conflation appears in Lemma A.18. Because Corollary 4.9 and Algorithm 2 invoke Eq. (9), the derivation of the variational method is formally invalid as written; the displayed Ψ_Λ in Eq. (10) uses the correct ∇_θ log q_θ covariance, so the formula may be salvageable, but the lemma and its proof must be corrected before the claim can be accepted.
  2. [§1 vs. §4.1, Algorithms 1–2] The introduction claims compatibility with any parameterised score model and names neural networks as an example, but this is not supported by the algorithms. Definition 4.4 and Eq. (7) evaluate q_θ directly, Algorithm 1 takes q_θ as input, and the variational Ψ_Λ in Corollaries 4.9 and 4.11 requires ∇_θ log q_θ. A general neural-network score s_θ without a known anti-derivative cannot be used, and no construction is supplied. The actual scope is explicit unnormalised-density/energy models of the form s_θ = ∇_x log q_θ; the abstract, introduction, and the final remark in Section 5 that the methods can be applied to NNs should be revised to state this restriction.
minor comments (5)
  1. [Assumption 4.2] The assumption is introduced as 'For any θ > 0', but θ is a parameter vector; it should read 'For any θ ∈ Θ'.
  2. [Theorem 4.6 and Theorem A.15] The main-text bound is a substantial simplification of Appendix Theorem A.15; the constants β_i and the additional β_3((n+r)/(nr))(...) term are not defined in Section 4.2.2. Please state that the displayed rate is a simplified corollary of A.15 and point explicitly to the full statement.
  3. [Figure 2] The caption says 'Average Fisher Divergence for Gaussian score estimates', but the experiment in Section 5.1.2 measures θ-error for the non-Gaussian ICA-style model; the caption should say 'Theta Error'.
  4. [Assumption A.13] In the displayed inequality, the right-hand side should compare f_{l,λ}(x,θ) with f_{l,λ}(x,θ'), not with f_{0,λ}(x,θ'); as written the Lipschitz condition is mis-stated.
  5. [Appendix C.2, Proposition C.5] The proof contains a set-inclusion in the wrong direction: it writes {|Z_r(θ)-Z(θ)|<ε} ⊆ ∪_k E_k(θ), but the argument needs the complementary inclusion for the large-deviation event; please correct this typo so the union bound is applied to {|Z_r(θ)-Z(θ)|>ε}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the marginal and truncated objectives are derived in-paper from standard score-matching integration-by-parts identities, and the sole overlapping-author citation (Liu et al. 2022) is an independently published theorem that is also re-derived in Appendix C.1.1.

full rationale

The paper's derivation chain is self-contained rather than circular. Proposition 4.3 obtains the marginal score matching objective LM(theta) from the marginal Fisher divergence by applying the standard Hyvärinen integration-by-parts identity to each marginal p_lambda; the proof in Appendix C.1 is explicit, and the step from matching all marginal scores to recovering the full density is an explicit identifiability assumption (Assumption 4.2(e)), not a hidden consequence of the objective. The truncated objective (5) and Proposition A.2 are re-derived with Green's theorem in Appendix C.1.1. Although Liu et al. (2022) is cited and shares an author with this paper, that citation is not load-bearing: the needed equivalence is written out in the appendix, and the cited paper is an independent JMLR publication, so it is real evidence rather than a circular authority. The IW estimator (6)-(7) is a standard importance-weighted Monte Carlo approximation of the marginal score, and the variational gradient in Corollary 4.9 follows by differentiating the already-established objective with the log-derivative trick; no parameter is fitted to a subset and then renamed a prediction. Theorem 4.6 is a uniform-convergence bound on the population marginal truncated Fisher divergence of the empirical minimiser, and Remark 4.7 explicitly cautions that this is not the full-data Fisher divergence, so the bound is honestly scoped. A separate concern that Lemma 4.8's equation (9) may misstate the theta-gradient covariance is a mathematical correctness issue, not a circularity, and it does not affect the importance-weighting central claim.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The ledger contains no invented physical entities. The free parameters are user-chosen hyperparameters or proposal distributions, not hidden constants fitted to force the result. The axioms are either standard score-matching regularity conditions or explicitly stated assumptions about missingness, identifiability, and the proposal or variational family.

free parameters (5)
  • IW proposal density p' = Isotropic Gaussian N(0, 16I) in experiments; user-chosen in general
    Equations (6) and (7) use an arbitrary proposal p' for importance sampling; its choice affects variance and requires support assumptions.
  • Importance sample count r = 10 in experiments
    Algorithm 1 uses r samples per data point. Theorem 4.6 needs r of order n for the best rate, but experiments use r=10.
  • Truncation weighting function g = Distance to boundary in Remark A.17
    The truncated objective in Equation (5) depends on an arbitrary weight function g satisfying boundary vanish conditions.
  • Variational conditional family p'_phi = Gaussian N(mu_phi(x_lambda), sigma^2 I) with a 2x200 neural network
    Used to approximate p_theta(.|x_lambda) for all observed subsets lambda; no misspecification analysis is provided.
  • Inner variational steps L = 10
    Algorithm 2 updates phi L times per theta step, chosen in Remark 4.14.
assumptions (7)
  • domain assumption Missing completely at random: M is independent of X
    Section 3.3 defines the core setting with M independent of X. The MNAR extension in Appendix A.1.4 instead assumes known probabilities phi_lambda.
  • domain assumption Identifiability: equality of all observed marginals implies equality of full densities
    Assumption 4.2(e) is required for Proposition 4.3 to conclude that the minimiser is the true unnormalised density.
  • ad hoc to paper The score model is the gradient of a known unnormalized density q_theta
    Algorithms 1 and 2 take q_theta as input, and the marginal score estimate in Equation (7) and the covariance term in Corollary 4.9 both require q_theta and log q_theta.
  • domain assumption Bounded likelihood ratio and bounded f_k in Assumption A.11
    The finite-sample bound for truncated IW requires q_theta/p' bounded above and below, plus bounded first and second derivatives.
  • domain assumption Sub-Gaussian Lipschitz conditions in Assumption A.13
    The covering-number argument in the proof of Theorem A.15 uses these sub-Gaussian constants to control the sup over theta.
  • standard math Standard regularity conditions for score matching integration by parts
    The paper invokes Hyvarinen (2005) for classical score matching and Liu et al. (2022) for truncated score matching, including Green's theorem boundary conditions.
  • domain assumption Support of X is contained in support of the proposal p'
    Proposition C.5 assumes supp(X) subset of supp(X') for the importance sampling estimate to be valid.

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Cite this review

Pith. "Pith review of Score Matching With Missing Data." pith.science (2026). https://pith.science/paper/GFAP6GWT

@misc{pith2026250600557,
  author       = {Pith},
  title        = {Pith review of: Score Matching With Missing Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFAP6GWT}},
  note         = {Machine review of arXiv:2506.00557}
}
read the original abstract

Score matching is a vital tool for learning the distribution of data with applications across many areas including diffusion processes, energy based modelling, and graphical model estimation. Despite all these applications, little work explores its use when data is incomplete. We address this by adapting score matching (and its major extensions) to work with missing data in a flexible setting where data can be partially missing over any subset of the coordinates. We provide two separate score matching variations for general use, an importance weighting (IW) approach, and a variational approach. We provide finite sample bounds for our IW approach in finite domain settings and show it to have especially strong performance in small sample lower dimensional cases. Complementing this, we show our variational approach to be strongest in more complex high-dimensional settings which we demonstrate on graphical model estimation tasks on both real and simulated data.

Figures

Figures reproduced from arXiv: 2506.00557 by the authors.

Figure 1
Figure 1. Average Fisher Divergence for Gaussian score [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Average Fisher Divergence for Gaussian score [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Mean AUC of star graph edge detection with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Mean AUC of dense graph edge detection with [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 7
Figure 7. Figure 7: Mean AUC of various methods when compared [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 6
Figure 6. Figure 6: Mean AUC of various methods when compared [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: Average parameter estimation error for truncated Gaussian score estimates alongside 95% confidence intervals [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Average Fisher Divergence with 95% C.I.s for various approaches. Lower is better. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Covariance and precision from a sample distribution from our normal experiment. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Marginalisation of the precision to remove dimension 1 by the naive approach (i.e. subsetting the precision) [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Average parameter estimation error (L2 norm) for ICA inspired model with 95% C.I.s for various methods. [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Covariance, precision, and marginalisations of the precisions to remove dimension 1 by the naive approach (i.e. [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Covariance, precision, and marginalisations of the precisions to remove dimension 1 by the naive approach (i.e. [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Mean AUC of various methods for edge detection of star-shaped GGM as we increase the dimension presented [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: ROC Curves in 4 separate runs for GGM estimation of a model truncated normal distribution with a star-shaped [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.