Ordering-based Causal Discovery via Generalized Score Matching

Dinh Phung; Edwin V. Bonilla; He Zhao; Trung Le; Vy Vo

REVIEW 4 major objections 4 minor 77 references

Ordering-based Causal Discovery via Generalized Score Matching

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A leaf node in a discrete causal model is the variable whose conditional probability given the rest is most random; peeling such leaves recovers the causal order.

desk verdict Novel discrete score-matching ordering idea with a real gap: both certification routes collapse, so Condition 2 is unverified and the variance measure used in experiments is not a randomness measure. read the letter →

arxiv 2601.16249 v3 pith:7PV22V3B submitted 2026-01-22 cs.LG cs.AI

Vy Vo , He Zhao , Trung Le , Edwin V. Bonilla , Dinh Phung This is my paper · ORCID

classification cs.LGcs.AI

keywords causaldiscoverydiscretedatascorematchingtopologicalorderleafdetectionrandomnessmeasuregeneralizedstructuralmodel

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

This paper extends score-matching causal discovery, previously limited to continuous additive-noise models, to discrete categorical data. It claims that when a randomness measure such as entropy never decreases along a causal edge, leaf nodes are exactly the variables maximizing the expected randomness of the reciprocal conditional probability p(X_i | X_-i). Peeling that leaf and repeating recovers a valid topological order without assuming additive noise or a particular functional form. The authors test the criterion on simulated and real-world networks and show that the recovered order improves existing structure-learning baselines on most settings. The load-bearing assumption is a monotonicity condition on randomness that the paper itself notes may fail on real data.

What carries the argument

The argument rides on two objects. The reciprocal discrete score function r_i(x_-i) := p(X_i | x_-i), the inverse of the generalized score for discrete distributions, turns leaf identification into a comparison of conditional distributions. The randomness measure φ — a symmetric, strictly concave function on probability vectors, e.g., entropy or negative variance — orders those distributions by spread. Condition 2 (non-decreasing randomness) requires φ(X_i | X_pa_i) ≤ φ(X_j | X_pa_j) for every edge i→j; Theorem 1 shows this monotonicity, together with the conditional-information inequality φ(X|Y) ≤ φ(X), makes the leaf's reciprocal score strictly maximal. A continuous-time discrete diffusion

What would settle it

Simulate a discrete SCM that explicitly enforces non-decreasing randomness on every edge and check whether the argmax criterion always selects a true leaf; a counterexample would refute Theorem 1. In parallel, recompute the asymptotic expansion in Proposition 1's proof (Appendix A.2, Eqs. 16–21) — the current derivation appears to contain an algebraic slip, so criterion (9) needs independent verification before Condition 2 can be certified from graph statistics alone.

Watch

Extended reading notes

Core claim

The central result is Theorem 1: in a discrete structural causal model obeying a 'non-decreasing randomness' condition, a node j is a leaf exactly when j = arg max_i E_{X_-i}[φ(r_i(X_-i))], where r_i(x_-i)=p(X_i|x_-i) is the reciprocal discrete score and φ is a randomness measure such as entropy or negative variance. Since every induced subgraph inherits the condition, repeating the argmax — peel the leaf, drop its column — yields a full topological order. The paper presents this as the first score-matching method for complete causal order from discrete data, subsuming non-decreasing-variance ANM conditions and generalizing entropic causality to multivariate order.

Load-bearing premise

Everything rests on Condition 2: the randomness of a child's conditional distribution must never be smaller than its parent's — a monotonicity the paper admits is likely violated on several real-world networks, and for which the proposed checkable proxy is not yet backed by a sound proof.

Editorial extensions

If this is right

Recovering a valid topological order from categorical observations requires no additive-noise or functional-form assumption; the discrete score alone suffices.
The recovered order can be fed to existing structure learners such as PC, GAM, and GES, improving F1 and SID on most simulated and real-world settings in the paper.
The non-decreasing randomness condition subsumes and generalizes earlier non-decreasing-noise-variance identifiability conditions for additive noise models.
When the condition holds only locally, the experiments suggest that partially correct sub-orders still carry enough signal to improve downstream DAG recovery.
The method stabilizes with roughly 500 samples, so it is usable in moderate-data regimes; remaining error is attributed more to assumption violation than to estimation noise.

Reading between the lines

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

Boundary test (inference): construct a chain A→B→C in which the child is a deterministic function of the parent, so its conditional randomness is lower rather than higher; the criterion would then select the wrong leaf, marking the precise boundary of the theory.
Independent check (inference): the asymptotic expansion in Proposition 1's proof (Appendix A.2, Eqs. 16–21) appears to contain an algebraic slip, so criterion (9) should not yet be treated as a certified test of Condition 2 until the derivation is repaired and re-verified.
Extension (inference): the theorem's proof machinery does not fundamentally depend on discreteness beyond the score definition, so a tractable estimator for singleton conditional densities could plausibly carry the same leaf criterion to continuous or mixed data.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper has a genuinely new idea — score-matching leaf ordering for discrete SCMs via concave randomness measures — but the two mechanisms it offers to actually use that idea are broken. Proposition 1's algebra is wrong, and the variance measure used in every simulation is not a valid randomness measure. The core theorem is a conditional statement that may still be true; it just hasn't been connected to practice.

What is new: extending the SCORE/CaPS line to discrete data with no additive-noise assumption, and generalizing entropic causality to arbitrary Schur-concave measures. That is a real contribution to a gap nobody else has filled. The writing is clear, the related-work table is helpful, and the experiments are broad (multiple synthetic families, six real networks, several pruning/insertion baselines). Code is released.

Where it wobbles: Condition 2 (non-decreasing randomness) is everything. If it holds, Theorem 1's leaf criterion follows from the conditional-information inequality. But the paper never gives a correct way to check Condition 2. In A.2, the expansion leading to Proposition 1 (Eqs. 19–21) has a wrong 1/α0 coefficient; the correct leading term is [Ki(ni−1) − Kj(nj−1)]/(2α0), not Ki(3ni+1)/2 − Kj(3nj+1)/2. So the checkable sufficient condition is unsupported. More seriously, Corollary 1 uses φVar = −Var, where Var(p) = Σ pk(log pk − μ)². This function is not Schur-convex (e.g., Var(0.99,0.01)=0.209 < Var(0.9,0.1)=0.435 despite majorization), so −Var is not concave and the inequality φ(X|Y) ≤ φ(X) — which the proof of Theorem 1 needs — fails for some distributions. A concrete counterexample is a 50/50 mixture of Bernoulli distributions with parameters 0.99 and 0.6. The consequence is that the measure driving all the empirical work falls outside the theorem. The paper also selects the randomness measure post hoc based on its own diagnostic, which is weaker still when the diagnostic is invalid.

The sequential subgraph argument is sketchier than the full-graph proof, but I think that one is patchable: when you remove a leaf, the remaining set is ancestral and the conditioning statements go through. Not a dealbreaker.

Bottom line: the conditional identifiability result is plausible and the empirical direction is worth attention, but the verification tools and the main experimental measure need real fixing. I would engage with this paper — send it to referees, expect a heavy revision — but I would not build a follow-up on it as is.

Referee Report

4 major / 4 minor

Summary. The paper proposes an ordering-based causal discovery method for discrete data. It defines a leaf-detection criterion based on the expected randomness, under a chosen measure φ, of the reciprocal singleton conditional p(X_i | X_{-i}). Theorem 1 states that, under a non-decreasing randomness condition along every directed edge, the leaf is exactly the variable maximizing E_{X_{-i}}[φ(p(X_i|X_{-i}))]; iterating this yields a topological order. The singleton conditionals are estimated with a continuous-time discrete diffusion model. The paper also provides a Dirichlet-based sufficient condition for the key assumption (Proposition 1), a variance-based instantiation (Corollary 1), a heuristic order-validity diagnosis, and experiments on synthetic and real-world discrete Bayesian networks, showing that the inferred orders can improve downstream PC/GES/GAM baselines.

Significance. If Theorem 1 is correct, the paper contributes a genuinely conditional identifiability result for full topological order recovery from discrete data without additive-noise assumptions, extending score-matching causal discovery to a setting where classical scores are undefined. The central derivation is structured as a clean conditional-information inequality with no fitted constants, and the authors provide code and extensive synthetic/real experiments. These are real strengths. However, two load-bearing supporting results are technically defective: Proposition 1's asymptotic expansion is wrong, and the variance functional used in Corollary 1 and in most experiments is not a valid randomness measure under the paper's own Definition 3. As a result, the claimed practical testability and the theoretical coverage of the empirical evaluation are currently unsupported. The central idea remains defensible, but the manuscript needs substantive revision.

major comments (4)

[App. A.2, Eqs. (16)–(21)] The asymptotic expansion in Proposition 1 is algebraically incorrect. Expanding Eq. (16) to leading order in 1/α0 gives H(X_i|X_pa_i) ≈ log n_i − (n_i−1)K_i/(2α0). Substituting into ΔH_{i,j} yields, at leading order, log(n_j/n_i) + [K_i(n_i−1) − K_j(n_j−1)]/(2α0). The paper instead produces the extra term [(1+n_i)K_i − (1+n_j)K_j]/α0 in Eq. (19), leading to c(n) = (3n+1)/2. The correct coefficient is (n−1)/2. Therefore inequality (9) is not derived from Eq. (16) and the stated sufficient condition for Condition 2 is unsupported. Since Proposition 1 is the only analytical certification of the key condition, this is a load-bearing error.
[§4, Corollary 1 and text before it] The variance functional Var(p)=Σ_k p_k(log p_k − μ)^2, μ=Σ_k p_k log p_k, is claimed to be Schur-convex because it is convex, and hence −Var is a valid randomness measure. This is false. For n=2, Var(0.99,0.01)=0.209 < Var(0.9,0.1)=0.435 although (0.99,0.01) majorizes (0.9,0.1), so Var is not Schur-convex. Consequently −Var is not Schur-concave and does not satisfy the conditional information inequality (10) that is essential to Theorem 1. A concrete failure: the equal mixture of (0.99,0.01) and (0.6,0.4) gives E[Var(X|Y)]=0.124 < Var(marginal)=0.2995, so E[−Var(X|Y)] > −Var(X). Thus Theorem 1 does not apply to Corollary 1. Since the experiments overwhelmingly use this variance measure, the reported experiments cannot be interpreted as validating the theorem.
[§6 and App. C.3] The paper explicitly admits in Sec. 6 that Condition 2 'is likely violated' on Sachs and Insurance, and that the experiments intentionally do not enforce any randomness measure or exact monotonicity. This is acceptable for a robustness study, but the paper also claims to address the testability gap. The proposed diagnosis (App. C.3) is explicitly heuristic and non-definitive, and Proposition 1 — the only analytic check — is invalid as shown above. Once Corollary 1 is also removed, no correct, checkable sufficient condition for Condition 2 remains, and no experimental configuration is guaranteed to be in the theorem's regime. The paper should provide a correct verifiable condition (e.g., a repaired Dirichlet analysis or a different parametric family) or clearly re-label the empirical results as heuristic demonstrations not covered by Theorem 1.
[App. A.1 and Algorithm 1] The proof of Theorem 1, as written, focuses on the full graph and then asserts the same argument works for any induced subgraph G[S]. The sequential leaf-peeling algorithm needs the criterion to hold on the remaining variables, where the conditioning set is X_{S\i}, not the original X_{-i}. The proof does not explicitly define pa_i, mb_i in the induced subgraph, nor verify that a leaf in G[S] satisfies E_{X_{S\l}}[φ(p(X_l|X_{S\l}))]=φ(X_l|X_pa_l) and that a non-leaf satisfies E_{X_{S\i}}[φ(p(X_i|X_{S\i}))]=φ(X_i|X_mb_i^S). This is likely repairable — parents of remaining nodes are unchanged and the Markov blanket in the induced subgraph contains the parents — but as written the sequential guarantee is not fully established. Please spell out the subgraph induction.

minor comments (4)

[Theorem 1 statement] The statement 'X_j is a leaf ⇔ j = arg max_i ...' presumes uniqueness of the arg max. If multiple leaves attain the same maximum, each leaf is an arg max but not necessarily the unique one. The intended statement is likely 'j is a leaf ⇔ j ∈ arg max_i ...'. Please clarify tie handling.
[§4, Proposition 1 setup] The notation in Eq. (8) is ambiguous: 'p(X_i | x_pa_i) ~ Dirichlet(α0/(n_i K_i) 1_{n_i})' should specify the vector length and the normalizing convention. Also, Condition 2 is stated with 'min_{j∈chi}' but what happens when ch_i=∅ is not stated explicitly in the definition; clarify that it applies only to nodes with children.
[§5, choice of randomness measure] The text says the randomness measure is selected 'yielding the least risk of violations based on the proposed diagnosis,' but the diagnosis is only described in App. C.3 and its results are not shown for this selection. Since the variance measure is theoretically invalid, the measure-selection procedure needs to be reported transparently and preferably repeated with a valid measure.
[Figure 1 and Tables] The main text reports only selected baselines and aggregated curves; error bars or standard deviations are not visible. Given the 10 random initializations, please add error bars to the figures or state that they were omitted for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a conditional derivation from an explicit monotonicity assumption, not an equation-level reduction to its inputs.

full rationale

The central result (Theorem 1, App. A.1) takes Condition 2 (non-decreasing randomness along edges) as an antecedent and derives that leaves maximize E[phi(p(X_i|X_-i))] using the conditional-information inequality (10) that follows from concavity (Definition 3). Condition 2 is not defined as the conclusion 'leaf = argmax'; it is a separate property of local CPTs, and the theorem's converse direction relies on the same inequality plus monotonicity. No fitted parameter is renamed as a prediction; no self-citation is load-bearing; no known result is merely relabeled. Concerns raised by the skeptical review are correctness/verifiability issues, not circularity: Sec. 6 explicitly concedes Condition 2 may be violated on real data; App. A.2 Eqs. (16)-(21) contain a questionable algebraic expansion for Proposition 1; and the claim in Sec. 4 (Corollary 1) that Var(p)=sum p_k(log p_k - mu)^2 is Schur-convex is contradicted by the binary example (0.99,0.01) vs (0.9,0.1), so -Var is not established as a valid randomness measure. These would undermine the applicability or proof of the theorem, but they do not make the derivation equivalent to its inputs. The per-dataset selection of variance vs entropy via the in-degree diagnosis is transparent model selection, not a constructed prediction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The central theorem is conditional on Condition 2; the only fitted ingredients in the method are the choice of randomness measure and the estimator hyperparameters. The diffusion estimator and the discrete score operator are adopted from prior literature.

free parameters (4)

Randomness measure φ = Variance for simulations, Insurance, Mildew; Shannon entropy for Earthquake, Sachs, Child, Alarm
Chosen per dataset based on the proposed in-degree diagnosis (§5), i.e., selected using the test data; the leaf criterion argmax E[φ] depends on this choice.
Diffusion estimator hyperparameters = 300 epochs, 3000 time steps, lr=0.001, hidden units=2d
Hand-chosen training configuration for the continuous-time discrete diffusion score estimator (App. B.1); accuracy of the estimated singleton conditionals, and hence the order, depends on these.
Independence-test threshold / pruning cutoffs = α=0.01 for χ² tests; cut-off 0.001 for GAM
Hand-chosen thresholds in the order-validity diagnosis and edge-pruning post-processing (Sec. 5, App. C.3); affects final DAG quality.
Dirichlet equivalent sample size α0 = large but unspecified
Parameter in Proposition 1's sufficient condition (8); the proposition only holds 'for sufficiently large α0', so the practical test requires choosing α0.

assumptions (7)

domain assumption Standard SCM conditions: Markov property, no latent confounders, independent exogenous variables with strictly positive density (Sec 2.1)
The factorization p(x)=∏ p(x_i|pa_i) and the Markov-blanket arguments in the proof of Theorem 1 (A.1) rely on these.
domain assumption Condition 1 (non-degeneracy): every conditional p(x_i|pa_i) is strictly positive
Needed so reciprocal scores r_i are well-defined and the conditional-information inequality is strict (A.1, Eq 13).
ad hoc to paper Condition 2 (non-decreasing randomness): φ(X_i|pa_i) ≤ φ(X_j|pa_j) for every edge i→j
The central identifiability assumption; Theorem 1 and the sequential leaf-peeling guarantee hold only under this monotonicity.
standard math Hickey's conditional information property: for a symmetric strictly concave φ, E[φ(X|Y)] ≤ φ(X), equality iff independence
Used in Eq (10) and throughout the proof of Theorem 1; also requires φ to be a valid measure of randomness (Definition 3).
standard math Completeness of the marginalization operator M (Brook 1964, Lyu 2012)
Justifies using singleton conditionals as a complete discrete 'score' (Sec 2.2).
domain assumption Consistency of the continuous-time discrete diffusion score estimator (Sun et al. 2022)
Algorithm 1 (B.1) assumes the trained network solves the ratio-matching objective (29) accurately enough to identify the argmax; no sample-complexity or sequential error-propagation guarantee is given.
domain assumption Faithfulness for the order-validity diagnosis (Algorithm 2)
The in-degree test treats each detected conditional dependence X_j ⊥̸⊥ X_i | S\{i} as a true edge; the paper notes 'conclusion is not definitive unless faithfulness is strongly assumed' (Sec 4, C.3).

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Pith. "Pith review of Ordering-based Causal Discovery via Generalized Score Matching." pith.science (2026). https://pith.science/paper/7PV22V3B

@misc{pith2026260116249,
  author       = {Pith},
  title        = {Pith review of: Ordering-based Causal Discovery via Generalized Score Matching},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PV22V3B}},
  note         = {Machine review of arXiv:2601.16249}
}

read the original abstract

Learning DAG structures from purely observational data remains a long-standing challenge across scientific domains. An emerging line of research leverages the score of the data distribution to initially identify a topological order of the underlying DAG via leaf node detection and subsequently performs edge pruning for graph recovery. This paper extends the score matching framework for causal discovery, which is originally designated for continuous data, and introduces a novel leaf discriminant criterion based on the discrete score function. Through simulated and real-world experiments, we demonstrate that our theory enables accurate inference of true causal orders from observed discrete data and the identified ordering can significantly boost the accuracy of existing causal discovery baselines on nearly all of the settings.

Figures

Figures reproduced from arXiv: 2601.16249 by the authors.

**Figure 1.** Experiments with (top) ER graphs of 2d degree, (bottom) real-world networks on 10, 000 samples. 6 Results and Discussion It is worth emphasizing that our simulations are deliberately designed to be noisy, without enforcing any specific randomness measure or exact monotonicity in the true causal order. Likewise, the real-world settings provide no guarantee that Condition 2 holds, and even there is a substantial chan… view at source ↗

**Figure 2.** Experiments on synthetic ER graphs of 4d degree. 5 10 15 20 d 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 SHD Dtop SCORE+PC PC SCORE+GAM GAM SCORE+BDeu GES-BDeu GES-BIC ORDCD 5 10 15 20 d 0.0 0.2 0.4 0.6 0.8 SID 5 10 15 20 d 0.0 0.2 0.4 0.6 0.8 1.0 F1 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Experiments on synthetic SF graphs of 2d degree. C.2 Sensitivity Analysis on Sample Size In Figures 5 to 11, we report the full experimental results on the real-world datasets where our methods are evaluated at varying sample sizes N ∈ {100, 500, 1000, 5000, 10000, 15000, 20000}. Meanwhile, the baseline performance is maximally fixed at N = 20000 for the ease of comparison. It can be seen that our performance begins… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Experiments on synthetic SF graphs of 4d degree. EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 SHD SCORE+PC PC SCORE+GAM GAM SCORE+BDeu GES-BDeu EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.0 0.2 0.4 0.6 0.8 1.0 SID EQ (5) S…

**Figure 5.** Figure 5: Experiments with real-world networks on 100 samples. EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 SHD SCORE+PC PC SCORE+GAM GAM SCORE+BDeu GES-BDeu EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.0 0.2 0.4 0.6 0.8 1.0 SID EQ (…

**Figure 6.** Figure 6: Experiments with real-world networks on 500 samples. C.3 Practical Diagnosis for Order Validity We now discuss a heuristic diagnosis for the correctness of a topological order estimated from an ordering-based causal discovery algorithm. The proposed approach mainly rel…

**Figure 7.** Figure 7: Experiments with real-world networks on 1, 000 samples. EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 SHD SCORE+PC PC SCORE+GAM GAM SCORE+BDeu GES-BDeu EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.0 0.2 0.4 0.6 0.8 1.0 SID E…

**Figure 8.** Figure 8: Experiments with real-world networks on 5, 000 samples. EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 SHD SCORE+PC PC SCORE+GAM GAM SCORE+BDeu GES-BDeu EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.0 0.2 0.4 0.6 0.8 1.0 SID E…

**Figure 9.** Figure 9: Experiments with real-world networks on 10, 000 samples. samples and rejecting the null hypothesis at a significance level of 0.01. Suppose the maximum in-degree of G is deg− max (i.e., every node has at most deg− max incoming edges), any node j with the estimated in-d…

**Figure 10.** Figure 10: Experiments with real-world networks on 15, 000 samples. EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 SHD SCORE+PC PC SCORE+GAM GAM SCORE+BDeu GES-BDeu EQ (5) SA (11) CH (20) IS (27) MD (35) AL (37) 0.0 0.2 0.4 0.6 0.8 1.0 SID…

**Figure 11.** Figure 11: Experiments with real-world networks on 20, 000 samples. the correlation between Dtop and the in-degree distributions estimated from the topological orders given the knowledge of deg− max = 3. We here normalize Dtop over the maximal number of edges in DAGs of d 2 to…

**Figure 12.** Figure 12: (a) Illustration of the correlation between Dtop and the in-degree distribution given deg− max = 3 across various graph degrees. (b) p-values of KCI tests across sample sizes and hypotheses of deg− max. Ordering-based Causal Discovery This family of methods typically …

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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