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REVIEW 4 major objections 4 minor 63 references

Score-based Generative Modeling for Conditional Independence Testing

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Score-based generative models can generate valid null-hypothesis samples for conditional independence testing, delivering asymptotic Type I error control and strong power.

desk verdict A well-executed diffusion-based CI test with strong experiments, but the headline Type I error theorem doesn't cover the implemented algorithm because the proof requires a sample split that Algorithm 1 does not perform. read the letter →

arxiv 2505.23309 v1 pith:VNVIL6KM submitted 2025-05-29 cs.LG cs.AI

Yixin Ren , Chenghou Jin , Yewei Xia , Li Ke , Longtao Huang , Hui Xue , Hao Zhang , Jihong Guan
show 1 more author
Shuigeng Zhou
This is my paper
classification cs.LGcs.AI MSC 62G1062G2062H15
keywords conditionalindependencetestingscore-basedgenerativemodelscorematchingLangevindynamicsTypeIerrorcontrolgoodness-of-fithigh-dimensionalstatistics
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

The paper sets out to show that score-based generative models can be turned into a conditional independence test. It trains a conditional score function by sliced score matching, uses Langevin dynamics to draw X values that preserve the X–Z relationship while breaking the X–Y link, and then compares the observed test statistic with the distribution of statistics from those generated null samples. The paper proves an asymptotic Type I error bound, and it adds a goodness-of-fit stage so a user can check whether the generated samples resemble the data. If correct, this gives a stable, high-dimensional-capable alternative to GAN-based conditional independence tests.

What carries the argument

The carrying object is the conditional score, $\nabla_x \log p(x\mid z)$, estimated by a conditional sliced score matching objective $J_\theta$ that avoids computing the trace of a Hessian. Langevin dynamics conditional sampling then updates $x^{(b)}_{(k+1)h} = x^{(b)}_{kh} + h\, s(x^{(b)}_{kh}, z; \theta) + \sqrt{2h}\, \xi_{kh}$, producing null triples that are conditionally independent of $Y$ given $Z$. Exchangeability of the resulting sequence, via the representation theorem for exchangeable sequences, is what turns a rank comparison of statistics into a valid $p$-value.

What would settle it

Simulate data from a true conditional density that is demonstrably outside the MLP family, for example a sharp multimodal mixture, set $H_0$ true, and run SGMCIT with large $n$; if the empirical Type I error at $\alpha = 0.05$ exceeds 0.05 by more than sampling error in many replicates, the bound in Theorem 5.4 does not hold under misspecification.

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Extended reading notes

Core claim

The central claim is that sampling from an estimated conditional score is enough to make conditional randomization tests valid in high dimensions. Under the null $X \perp\!\!\perp Y \mid Z$, the generated triples $(X^{(b)}, Y, Z)$ are exchangeable with the observed triple when the score model is accurate; the paper proves that the total-variation error of the generated conditional distribution vanishes in probability as $n$ grows (Theorem 5.3), and therefore the Type I error satisfies $P(p\text{-value} \le \alpha \mid H_0) \le \alpha + o_p(1)$ (Theorem 5.4). The same machinery yields an interpretable two-sample goodness-of-fit check of the generated samples. The paper's experiments report Type I error near the nominal level with high power across benchmark, high-dimensional confounder, and chain-structured settings.

Load-bearing premise

Everything rests on Assumption 4: the true joint density must be exactly inside the parametric family $\{p(x,z;\theta)\}$ that the MLP score model represents, and no argument shows that real conditional densities are.

Editorial extensions

If this is right

  • Under $H_0$ and Assumptions 1–10, the $p$-value exceeds $\alpha$ with probability at most $\alpha + o_p(1)$, so large-sample Type I error control holds.
  • Any chosen statistic $\rho$ can be plugged in; exchangeability of generated triples makes the rank-based $p$-value valid whenever the generated distribution matches the true conditional.
  • The goodness-of-fit stage lets a practitioner detect when generated samples are not trustworthy enough for the test result to be reliable.
  • In the reported experiments, SGMCIT holds Type I error near 0.05 while GAN-based GCIT and DGCIT inflate it, and it maintains power in the 100-dimensional confounder setting.
  • The method is computationally light compared with kernel-based tests at large sample sizes, according to the runtime measurements reported in the paper.

Reading between the lines

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

  • One implicit extension is that the same conditional score estimator could serve any conditional randomization test, not only the specific rank statistics used here.
  • The goodness-of-fit stage is a practical safety check, but it cannot repair a misspecified score family; a nonparametric or noise-perturbed score estimator might relax the well-specified assumption.
  • A concrete testable extension: apply SGMCIT to a chain-structured causal graph with nonlinear conditional densities to see whether Type I error remains controlled when the MLP family is far from the truth.
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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

4 major / 4 minor

Summary. The paper proposes SGMCIT, a conditional independence testing method that trains a sliced conditional score-matching model on (X,Z) pairs, generates null-hypothesis samples of X given Z via Langevin dynamics, checks the generated samples with a goodness-of-fit stage, and computes a Monte Carlo p-value by comparing the statistic on observed triples with statistics on generated triples. The authors claim asymptotic Type I error control (Theorem 5.4) and strong power, supported by score-consistency and total-variation error bounds, and they report experiments on synthetic and real datasets showing favorable performance against several existing CI tests, including GAN-based methods and a concurrent diffusion-based method.

Significance. If the theoretical guarantees held for the implemented algorithm, the paper would make a useful contribution: it would provide a generative-model-based CI test with a principled score-matching training objective, an interpretable goodness-of-fit check, and a formal asymptotic Type I error bound, while avoiding GAN training instability. The paper also provides code and compares against a broad set of baselines, which is a practical strength. However, the central validity claim currently has a load-bearing gap between the theory and the algorithm: the proof requires a train/test split that Algorithm 1 does not perform, and the parametric identifiability assumption is not satisfied by the implemented MLP score model. These issues must be resolved before the claimed guarantees can be considered established.

major comments (4)
  1. [§5, Theorem 5.4; Appendix E; Algorithm 1] The proof of the Type I error bound assumes that the score model is trained on a separate sample of size N and the CI test is run on an independent sample of size n (Appendix E, first paragraph: "we separate the samples used in the previous stage of generative modeling from the samples used in the CI test"). Algorithm 1, however, trains the score model on the full dataset D (Line 2) and computes the p-value on the same D (Line 9). Consequently, the generated triples (X^(b),Y,Z) depend on the observed X through the estimated parameter theta-hat, so the conditional independence used in Proposition D.1 and Lemma E.1 does not hold. The exchangeability argument requires the score function to be fixed before seeing the test data; when theta-hat is fitted to D, the map from initialization noise to X^(b) is a function of X, and the representation-theorem step in Proposition D.1 no longer applies. This gap is load-bearing: Theorem 5.4's conclusion P(p-value <= alpha | H0) <= alpha + o_p(1) is not a valid consequence for SGMCIT as implemented. The authors should either modify Algorithm 1 to use a sample split (train on a subset, test on the remaining data) and state the theorem for that version, or provide a different argument that justifies the exchangeability of the implemented procedure.
  2. [Assumption 4 and Theorem 5.2; §6/F.2] Assumption 4 requires the parametric family {p(x,z;theta)} to be well-specified, i.e., the true joint density equals p(x,z;theta*) for some theta*. The actual score model is a three-layer MLP with 64 hidden units (Appendix F.2), and no argument is given that the true conditional densities lie in this family. Under misspecification, Lemma 5.1's equivalence L_theta = 0 iff theta = theta* and the consistency Theorem 5.2 collapse, and the total-variation bound of Theorem 5.3 and the Type I error bound of Theorem 5.4 inherit the failure. Even if the practitioner interprets the MLP as a flexible approximation rather than an exact parametric model, the stated theorems do not cover that case. The paper should either prove the results under a misspecification-aware assumption (e.g., bounding the approximation error in the score or in TV distance), or explicitly restrict the theoretical claims to cases where the model class is exactly specified and clarify that the experiments do not verify this condition.
  3. [Theorem C.2 and Eq. (34)] The proof of Theorem C.2 (and hence Theorem 5.3) contains a probabilistic step that is not justified as written. From the pointwise convergence s(x,z;theta-hat) -> s(x,z;theta*) in probability, Eq. (34) asserts that the event E_{x~p(x|z)}[||s(x,z;theta*) - s(x,z;theta-hat)||^2] <= eps_c^2 has probability tending to 1. Pointwise convergence does not imply convergence of the L2 expectation without additional uniform integrability or dominated-convergence-type conditions, which are neither stated nor proved. This step is needed to conclude that the score estimation error satisfies Assumption C.3 asymptotically, and the gap should be repaired with a precise regularity condition (e.g., uniform integrability of the squared score error under p(x|z)) or a different argument based on the actual convergence mode established in Theorem 5.2.
  4. [§4.2, Algorithm 1, Lines 5-6] The goodness-of-fit stage is described as checking whether the generated samples are 'good enough', but no formal decision rule is specified, and the p-value in Eq. (10) is computed regardless of the GOF outcome. Since the GOF stage is presented as a key practical safeguard for the validity of the CI test, the paper should define a precise rule (e.g., reject the CI test as unreliable if an MMD-based p-value is below a threshold), and state what guarantee, if any, the GOF stage provides when it is used as a filtering step. Without a formal rule, the claim that the GOF stage 'ensures the validity of generated samples' is not supported.
minor comments (4)
  1. [Throughout] There are several typos and notational inconsistencies: 'Exchangeablility' in the title of Proposition 4.1 and Appendix D; 'signifiance level' in Appendix E; 'squre' in several figure labels; 'CDCCIT' instead of 'CDCIT' in Figure 17; and the notation [B] is used before it is defined in the main text (it is defined only in a footnote and in Appendix A). The paper should be carefully proofread.
  2. [Theorem 5.4 and Appendix E] Theorem 5.4 states the Type I error bound 'as n -> infinity', but the proof in Appendix E uses N for the training sample size and n for the test sample size and concludes convergence 'as N -> infinity'. The limit variable is inconsistent with the theorem statement, and this is not merely cosmetic because Algorithm 1 uses the same data for both roles. The authors should align the notation and state clearly which sample size drives the asymptotics in the implemented version.
  3. [§4.1 and Fig. 1] The paper does not specify the value of B, the number of generated null-sample sets, in the main text or in Appendix F. Since the p-value resolution and the exchangeability argument depend on B, this hyperparameter should be reported for reproducibility.
  4. [Appendix E, Eq. (47)] The derivation of the total-variation term uses the phrase 'based on the property that (x, x^(1), ..., x^(B)) is independent of each other'; this is imprecise. The independence should refer to the conditional independence of the vectors given (Y,Z), and the equality should be stated with the appropriate conditional distributions. Clarifying this would improve readability without changing the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Type I error bound is assembled from external score-matching and Langevin convergence results plus a standard exchangeability/rank argument, and no fitted quantity is renamed as a prediction.

full rationale

The derivation chain is self-contained rather than circular. Lemma 5.1 and Theorem 5.2 import sliced-score-matching consistency from Song et al. [49]; Theorem 5.3 imports Langevin/log-Sobolev convergence from Lee et al. [23]; Proposition D.3 is the standard exchangeability/rank bound for p-values. None of these citations is by the present authors, and none assumes the target Type I error conclusion. The goodness-of-fit stage is a heuristic diagnostic and is not used to define the p-value in Eq. (10). The only in-scope concern is a proof-implementation mismatch: Appendix E proves the Type I error bound under a train/test split ('we separate the samples used in the previous stage of generative modeling from the samples used in the CI test'), whereas Algorithm 1 trains the score model on the full dataset D (line 2) and computes rho(D) on the same D (line 9). This means the exchangeability premise of Proposition D.1 is not guaranteed for the implemented procedure, so Theorem 5.4's conclusion does not follow for Algorithm 1 as written. That is a validity/correctness gap, not circularity: the theorem's conclusion is not assumed in its premises, and no fitted parameter is renamed as a prediction. The authors' self-citations [33-35] appear only as related work and are not load-bearing.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard score-matching assumptions, strong model-family assumptions including well-specification, positivity, smoothness, and log-Sobolev inequalities, and an unstated train and test separation for the p-value exchangeability. No invented physical or mathematical entities are introduced. Practical hyperparameters h, T, m, and B are chosen by hand rather than derived from the theorem's constants.

free parameters (4)
  • Langevin step size h = 0.1
    Chosen by hand; the theorem requires h = epsilon^2 / (2720 d_x L^2 C_LS), which depends on unknown constants that are not estimated.
  • Langevin time horizon T (number of steps) = 200 steps
    Chosen by hand; the theorem requires T = 4 C_LS ln(2K/epsilon^2), so the fixed value may not satisfy the TV bound for all conditional distributions.
  • Projection count m = 1
    Chosen to balance variance and computational cost; the paper states m = 1 is already good but gives no theoretical justification.
  • Number of null sample sets B = not reported in main text
    Algorithm input that sets p-value granularity and cost; the paper does not state the value used in experiments.
assumptions (8)
  • domain assumption Assumption 4: the model family {p(x,z;theta)} is well-specified, so p(x,z) = p(x,z;theta*) for some theta*.
    Needed for Lemma 5.1 and Theorem 5.2; the actual three-layer MLP is not shown to contain the true density.
  • domain assumption Assumption 5: p(x,z;theta) > 0 for all theta and all (x,z).
    Used in Lemma 5.1 and Lemma B.2 to convert score equality into density equality.
  • domain assumption Assumption 9: log p(x|z) is C^1 and L_z-smooth in x with L_z >= 1.
    Required for the Langevin dynamics TV bound in Theorem 5.3 and Theorem C.1.
  • domain assumption Assumption 10: each p(x|z) satisfies a log-Sobolev inequality with constant C_{z,LS} >= 1.
    Required for polynomial convergence of Langevin dynamics; multimodal or heavy-tailed conditionals may fail this.
  • domain assumption Assumption C.3: the score estimate is L2-accurate, E_x[||grad log p(x|z) - s(x,z;theta_hat)||^2] <= epsilon_z^2.
    Used in Theorem C.1; Theorem C.2 attempts to derive it from pointwise consistency, but the proof relies on uniform convergence that is not fully established.
  • ad hoc to paper Training data for the score model is independent of the samples used to compute the test statistic.
    Appendix E explicitly separates N training and n test samples, but Algorithm 1 and the synthetic experiments do not state such a split; without it, Proposition 4.1 exchangeability is not justified.
  • standard math Diaconis-Freedman representation theorem for finite exchangeable sequences.
    Used in Proposition 4.1 and Proposition D.1 to conclude exchangeability from conditionally i.i.d. generation.
  • standard math Lee et al. [23] convergence result for score-based generative models with L2-accurate scores.
    Imported as the core error bound for the Langevin sampling stage in Theorem C.1.

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Cite this review

Pith. "Pith review of Score-based Generative Modeling for Conditional Independence Testing." pith.science (2026). https://pith.science/paper/VNVIL6KM

@misc{pith2026250523309,
  author       = {Pith},
  title        = {Pith review of: Score-based Generative Modeling for Conditional Independence Testing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNVIL6KM}},
  note         = {Machine review of arXiv:2505.23309}
}
read the original abstract

Determining conditional independence (CI) relationships between random variables is a fundamental yet challenging task in machine learning and statistics, especially in high-dimensional settings. Existing generative model-based CI testing methods, such as those utilizing generative adversarial networks (GANs), often struggle with undesirable modeling of conditional distributions and training instability, resulting in subpar performance. To address these issues, we propose a novel CI testing method via score-based generative modeling, which achieves precise Type I error control and strong testing power. Concretely, we first employ a sliced conditional score matching scheme to accurately estimate conditional score and use Langevin dynamics conditional sampling to generate null hypothesis samples, ensuring precise Type I error control. Then, we incorporate a goodness-of-fit stage into the method to verify generated samples and enhance interpretability in practice. We theoretically establish the error bound of conditional distributions modeled by score-based generative models and prove the validity of our CI tests. Extensive experiments on both synthetic and real-world datasets show that our method significantly outperforms existing state-of-the-art methods, providing a promising way to revitalize generative model-based CI testing.

Figures

Figures reproduced from arXiv: 2505.23309 by the authors.

Figure 1
Figure 1. The framework of conditional independence testing with score-based generative modeling. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Left: The visualization results for the marginal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Results of conditional independence tests on benchmark datasets. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (13 more)
Figure 5
Figure 5. Figure 5: Left: Results in the high-dimensional confounder [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: CI testing results on real data. Left: Visualization [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: The visualization results of SGMCIT for the marginal distribution of [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: The visualization results of GCIT for the marginal distribution of [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: The visualization results of DGCIT for the marginal distribution of [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: The visualization results of GCIT, DGCIT and SGMCIT for the marginal distribution of [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Additional results of conditional independence tests for Cases 1 and 2 on benchmark datasets. [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Additional results of conditional independence tests for Cases 3 and 4 on benchmark datasets. [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Results of running time. Left: The results w.r.t. the sample size. Right: The results w.r.t. the dimension of [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Additional results of CI tests. (a) Additional results of CI tests for Cases 1 and 2 on benchmark datasets. (b) Additional [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: The visualization results of SGMCIT and CDCIT for the distribution of [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: Visualization results of SGMCIT and CDCIT on real data with 50000 sample size. [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: Results of running time of CDCIT and SGMCIT (Ours). Left: The results w.r.t. the sample size. Right: The results w.r.t. [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]

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Pith tools

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