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Distinguishing Cause from Effect with Causal Velocity Models

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper establishes a noise-agnostic score identity for bivariate causal direction: a joint distribution is generated by an SCM with causal velocity $v$ if and only if $s_x(x,y)=-\partial_y v(y,x)-v(y,x)s_y(x,y)+s_x(x)$, and comparing…

desk verdict The velocity–score identity is the real contribution; the direction-selection rule is a heuristic that needs either a separation proof or a humbler framing. read the letter →

arxiv 2502.05122 v2 pith:ZEYO4YCK submitted 2025-02-07 stat.ML cs.LGstat.ME

Johnny Xi , Hugh Dance , Peter Orbanz , Benjamin Bloem-Reddy This is my paper · ORCID
classification stat.MLcs.LGstat.ME
keywords causaldiscoverybivariatestructuralmodelsscorefunctiondynamicalsystemsmeasuretransportcontinuityequationnoise-agnosticinferencecounterfactualcurves
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 tries to establish that, for bivariate systems, the causal direction can be read from the score function of the observed data if the generative mechanism is re-expressed as a dynamical system. The central move is to treat the cause variable as time and describe the SCM's counterfactual curves through a causal velocity. A score identity connects that velocity to the joint and marginal scores without ever specifying the noise distribution. The paper then compares a goodness-of-fit objective built from the identity in the $X\to Y$ and $Y\to X$ directions, and whichever direction fits is inferred to be causal. This matters because the velocity formulation covers mechanisms that additive and location-scale noise models cannot, while still allowing simple parametric velocity families to be estimated directly from data.

What carries the argument

The central object is the causal velocity $v(y,x)$, the derivative at the factual point of the counterfactual curve that maps an observed $y$ to its value had the cause $x$ been changed infinitesimally. It plays the role of the velocity field in an ODE whose time direction is the cause variable, generated by the flow $\varphi_{x,x'}(y)=f_{x'}(f_x^{-1}(y))$. The paper pairs this velocity with the continuity equation $\partial_x \log p(y\mid x) = -\partial_y v(y,x) - v(y,x)\partial_y \log p(y\mid x)$, converting it into the score identity of Theorem 4.1; this conversion is the mechanism that turns observational scores into a noise-agnostic regression target.

What would settle it

Compute the true joint and marginal scores for a generated SCM outside the ANM and LSNM classes and fit the velocity in the anti-causal direction using those exact scores; if the anti-causal fit reaches zero or matches the causal fit, the direction-selection claim fails. A concrete variant is to choose a velocity whose flow has closed-form counterfactuals, generate the exact joint density, and then check whether any alternative velocity in the same family satisfies the Theorem 4.1 identity in the reverse direction.

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Extended reading notes

Core claim

The paper's central claim is an if-and-only-if identity. Under regularity conditions, a joint distribution with full support and differentiable log-densities can be represented by a structural causal model with velocity $v$ exactly when $$s_x(x,y) = -\partial_y v(y,x) - v(y,x)\,s_y(x,y) + s_x(x)$$ holds for all $(x,y)$, where $s_x(x,y)$ and $s_y(x,y)$ are partial derivatives of the joint log-density and $s_x(x)$ is the marginal score of the cause. The identity is the log form of the continuity equation applied to the SCM flow, and it makes the noise distribution disappear from the estimation problem. The paper fits $v$ by minimizing $L(v)=\mathbb{E}[(s_x(X)-\partial_y v(Y,X)-s_v(X,Y))^2]$, where $s_v(x,y)=s_x(x,y)+v(y,x)s_y(x,y)$ is the directional derivative of the log-density along the causal curve, and chooses the direction with the smaller fitted objective. Because nothing in the objective requires an additive or location-scale mechanism, the same procedure applies to a wider class of models and needs no assumption on the noise.

Load-bearing premise

The load-bearing premise is that the joint and marginal score functions of the observed distribution can be estimated accurately enough from finite data; the paper's own ablations show discovery quality degrades when score estimates are poor, and discrete or integer-valued benchmark instances cannot be used because their scores are not defined.

Editorial extensions

If this is right

  • For any mechanism whose velocity can be represented, causal direction can be decided without fitting an additive-noise or location-scale model and without Gaussian assumptions on the noise.
  • The same goodness-of-fit value can flag model misspecification or non-identifiability when the score is estimated well, because the identity holds only when the velocity matches the data-generating SCM.
  • Integrating the estimated velocity recovers counterfactual curves, so velocity estimation yields a counterfactual model from observational data as a by-product.
  • If the score estimators converge, the fitted objective converges at a rate no worse than the score estimators, so improvements in nonparametric score estimation directly improve the reliability of the causal discovery procedure.

Reading between the lines

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

  • Applied to multivariate data, the pairwise score comparison could in principle be combined with graph-structure search, since the identity is local and does not depend on the noise distribution; the paper itself leaves that extension unexplored.
  • The identifiability equation obtained by writing the continuity equation in both directions suggests that similar uniform non-identifiability criteria might be derivable for other mechanism classes beyond ANM and LSNM, not just case-by-case checks.
  • Because the method needs a well-defined log-density, extending it to discrete or integer-valued variables would require smoothed densities or surrogate scores; the paper explicitly drops such benchmark instances rather than handling them.
  • The velocity parametrization also gives practitioners a way to encode mechanistic knowledge directly as infinitesimal-intervention responses, so a domain-specified velocity could be tested against data before making any parametric noise assumption.
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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

3 major / 5 minor

Summary. The paper introduces a parametrization of bivariate structural causal models (SCMs) through a “causal velocity” v(y,x), viewing the cause variable as time in a dynamical system. The central theoretical result (Theorem 4.1) is an identity relating the joint score, marginal score, and velocity via the continuity equation: sx(x,y) = -∂y v(y,x) - v(y,x) sy(x,y) + sx(x), holding if and only if the joint distribution is representable by an SCM with that velocity. The paper proposes a goodness-of-fit objective L(v) based on this identity, estimates scores nonparametrically, fits velocities in both candidate directions, and selects the direction with smaller objective. Theorem 5.1 states a consistency result for the empirical objective at a fixed v, and Proposition 5.2 derives a PDE condition for non-identifiability. Experiments on synthetic and benchmark data show the method succeeding on Velocity and Sigmoid benchmarks where ANM/LSNM methods struggle, with ablation studies examining the dependence on score estimation quality.

Significance. The velocity-based parametrization is a conceptually new and elegant way to connect SCMs, counterfactuals, and score functions, extending functional causal discovery beyond the ANM and LSNM classes without specifying noise distributions. The paper is honest about the central role of score estimation, provides code, and includes oracle-score experiments that cleanly isolate the behavior of the objective. If the method is reliable, it offers a practical tool for bivariate causal discovery in settings where existing methods are misspecified. However, the theoretical support for the direction-selection rule is incomplete, and the empirical results show non-negligible error rates even in well-specified settings, which tempers the practical significance until the decision rule is better understood.

major comments (3)
  1. [Section 5, Eq. (15)–(18)] The proposed causal discovery rule selects the direction with the smaller empirical objective, but the paper provides no population-level separation guarantee for the velocity classes actually used (B-LIN, B-QUAD, V-NN). Theorem 4.1 characterizes representability in a single direction, and Theorem 5.1 only bounds |That L_n(v) - L(v)| for a fixed v; neither controls the gap between the two directional infima. Since every smooth full-support bivariate density is generated by some bijective SCM in both directions, all identifiability must come from restricting the velocity class, and Section 5.2 explicitly stops short of a uniform identifiability analysis, stating that a general analysis “may require new techniques.” This is load-bearing for the central claim that the method distinguishes cause from effect.
  2. [Appendix B.3 (proof of Theorem 5.1)] The consistency proof bounds the empirical-process term by Rademacher averages over the class {f(s,·) : s ∈ B(H,M)}. In the expansion (36), product classes such as B(H1)⊗B(H2) and B(H1)⊗B(H3)⊗{h2} are treated as closed balls in tensor-product RKHSs. The set of products of two functions with bounded RKHS norm is not itself a ball in the tensor-product RKHS, and the assertion that “every term in (36) is the Rademacher average of a closed ball in an RKHS of bounded functions” is not justified. Thus Theorem 5.1, which is presented as establishing consistency of the estimation procedure, is not fully supported by the supplied proof.
  3. [Section 5.2, Proposition 5.2] Proposition 5.2 is stated as an “if and only if” characterization of non-identifiability, but the proof in Appendix C only equates the two mixed partial derivatives (50) and (51). Sufficiency—that equality of these derivatives implies existence of a joint density satisfying both directional continuity equations—is not shown. Moreover, the proposition does not yield a measure-zero or lower-bound argument for the specific basis classes (B-LIN, B-QUAD) used in the experiments, so the identifying power of those classes is only empirically demonstrated, not theoretically established.
minor comments (5)
  1. [Section 3, Definition 3.1] The sentence “It is easy to see that Equation (9) satisfies the axioms of a flow Theorem 2.1” is missing a word; it should read “of a flow in Theorem 2.1.”
  2. [Section 7.2] The phrase “which we suspect is due to the score being well-estimated” is informal; a more precise statement or a reference to the score-MSE results would strengthen the explanation.
  3. [Section 7.3, Table 8] For the well-specified LSNM experiment, the success rate is only 73% at n=5000 and 80% at n=10000 even when the true score is not used; the text should explicitly discuss why a well-specified model with improving score estimates still does not approach perfect accuracy.
  4. [Section 7.1 / Table 4] The Sigmoid benchmark is described verbally as “a variation on LSNMs with additional post-nonlinear and affine transformations,” but the mechanism in Table 4 is sufficiently complex that a brief interpretation or a reference to the table in the main text would improve reproducibility.
  5. [Figure 5 caption] The caption states “Upper and lower limits indicate 1st– 3rd quartile over 100 datasets,” but the figure appears to show shaded bands; specifying whether the limits are bands or error bars would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the score–velocity identity is derived from the continuity equation, and the causal-direction rule is a goodness-of-fit comparison rather than a fitted restatement of the target.

full rationale

The central result, Theorem 4.1 (Eq. 14), is derived from the log-continuity equation (Eq. 13), which the paper obtains from standard flow/transport results (Theorem A.2, from Santambrogio and Arnold). The 'if and only if' claim is supported by the uniqueness of continuity-equation solutions for a fixed velocity and by the flow construction in Theorem 3.2. No parameter is fitted to the causal-direction label. The objective L(v) in Eq. 15 is the squared residual of the identity, and causal discovery compares this residual under a restricted velocity class in each direction. That is a goodness-of-fit procedure, not a re-derivation of the conclusion from assumptions that already contain it. The paper explicitly acknowledges the main limitation: Section 5.2 says a general identifiability analysis 'may require new techniques,' and Theorem 5.1 bounds |hat L_n(v) - L(v)| for fixed v rather than the argmin gap. These are real methodological gaps for the direction-selection rule, but they are not circularity. The only self-citation (Dance & Bloem-Reddy 2024, cocycles) is contextual and is not used to prove Theorem 3.2 or Theorem 4.1; those proofs rely on Arnold and Santambrogio. The paper is therefore self-contained for its main derivation, with the caveat that population-level separation of the two directional losses is not established.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The method relies on standard dynamical systems theory and score estimation results. The main free parameters are the velocity model parameters and score estimator hyperparameters. No new physical or conceptual entities are introduced.

free parameters (2)
  • Velocity model parameters (basis coefficients or MLP weights) = Varies by dataset
    The velocity is estimated by minimizing the goodness-of-fit objective; these parameters are the model's free parameters.
  • Score estimator bandwidth and regularization = Median heuristic for Stein; Silverman rule for KDE; lambda=0.1 for Stein
    These hyperparameters control score estimation and affect the downstream objective; they are chosen by heuristics or fixed constants.
assumptions (4)
  • standard math Continuity equation solution uniqueness (Santambrogio 2015, Thm 4.4)
    Used to prove Theorem 4.1 and to justify that the score identity characterizes SCMs with a given velocity.
  • domain assumption SCM is bijective with differentiability and full support (Section B.1)
    The velocity parametrization and the counterfactual interpretation require f_x bijective and differentiable, and P_{X,Y} absolutely continuous with full support.
  • domain assumption Score estimators converge in RKHS norm at rate n^{-1/alpha} (Theorem 5.1)
    The consistency result for the objective estimator assumes this convergence, which is not verified for all practical settings.
  • domain assumption Identifiability: the causal direction yields a strictly smaller limit objective than the anticausal direction for the chosen velocity model class
    The method assumes this to choose direction; it is only characterized via a PDE, not proven for the proposed basis families.

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Cite this review

Pith. "Pith review of Distinguishing Cause from Effect with Causal Velocity Models." pith.science (2026). https://pith.science/paper/ZEYO4YCK

@misc{pith2026250205122,
  author       = {Pith},
  title        = {Pith review of: Distinguishing Cause from Effect with Causal Velocity Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZEYO4YCK}},
  note         = {Machine review of arXiv:2502.05122}
}
read the original abstract

Bivariate structural causal models (SCM) are often used to infer causal direction by examining their goodness-of-fit under restricted model classes. In this paper, we describe a parametrization of bivariate SCMs in terms of a causal velocity by viewing the cause variable as time in a dynamical system. The velocity implicitly defines counterfactual curves via the solution of initial value problems where the observation specifies the initial condition. Using tools from measure transport, we obtain a unique correspondence between SCMs and the score function of the generated distribution via its causal velocity. Based on this, we derive an objective function that directly regresses the velocity against the score function, the latter of which can be estimated non-parametrically from observational data. We use this to develop a method for bivariate causal discovery that extends beyond known model classes such as additive or location scale noise, and that requires no assumptions on the noise distributions. When the score is estimated well, the objective is also useful for detecting model non-identifiability and misspecification. We present positive results in simulation and benchmark experiments where many existing methods fail, and perform ablation studies to examine the method's sensitivity to accurate score estimation.

Figures

Figures reproduced from arXiv: 2502.05122 by the authors.

Figure 1
Figure 1. Given a SCM X → Y and an observation (x, y), the causal curve represents the implied counterfactual outcomes had x been x ′ (top). The derivative of the causal curve, which we call the causal velocity function v(y, x), generates all the causal curves and hence characterizes the counterfactuals of the SCM (bottom). In this paper, we show how the velocity can be estimated directly from data using the score function, w… view at source ↗
Figure 2
Figure 2. Distributions of causal curves obtained by sampling 100 initial conditions from p(y | x = x0) = N (0, 1). Note here the ANM and LSNM cross-sections p(y | x) are necessarily Gaussian also, no matter how flexible the underlying function classes are. On the other hand, the exponential velocity yields non-Gaussian conditionals at any x ̸= 0, exhibiting right and left skew as x < 0 and x > 0, respectively. the log-densit… view at source ↗
Figure 3
Figure 3. If the model is well-specified and the true score is given, velocity functions as obtained by minimizing Equation (15) (up to Monte Carlo approximation error) can be integrated numerically (left) to recover the true mechanism (right). Observe that (14) is entirely in terms of observable scores, and thus is agnostic to the unobserved noise distribution Pϵy . Indeed, SCMs with the same mechanism but different noise di… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: If the ground truth score is known, the anticausal GoF is orders of magnitude larger than the causal GoF. In this case, cause can be distinguished from effect with certainty even with as few as 100 samples. Right: As the score estimate quality improves with larger samp…
Figure 5
Figure 5. Figure 5: Score estimation and causal discovery performance as sample size increases in a well-specified ANM benchmark. Upper and lower limits indicate 1st–3rd quartile over 100 datasets. The larger error at n = 10000 is likely due to an increased chance of encountering outliers…
Figure 6
Figure 6. Figure 6: Example datasets from each synthetic benchmark. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: The estimated causal curves better resemble the ground truth as the score estimation improves in a well-specified LSNM. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

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