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arxiv: 2509.12981 · v2 · submitted 2025-09-16 · 💻 cs.LG · stat.ML

Causal Discovery via Quantile Partial Effect

Yikang Chen , Xingzhe Sun , Dehui Du This is my paper

Pith reviewed 2026-05-18 15:48 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords causal discoveryquantile partial effectidentifiabilityobservational distributioncausal directionfisher informationbasis function test
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The pith

Assuming the quantile partial effect of cause on effect lies in a finite linear span makes the causal direction identifiable from the observational distribution alone.

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

The paper demonstrates that if the quantile partial effect of one variable on another fits inside a finite linear span, the direction of causation can be recovered solely from the joint observational distribution of the two variables. This result generalizes earlier identifiability theorems that required additive noise, heteroscedastic noise, or other restrictions inside functional causal models. Because the quantile partial effect is computed directly from the data, the approach does not invoke hidden mechanisms, noise distributions, or the Markov condition. Instead it exploits visible asymmetries in the shape of the observed distribution. Readers should care because the method supplies concrete tests, via basis expansions for pairs and Fisher information for larger sets, that have been checked on many bivariate and multivariate datasets.

Core claim

When the QPE of cause on effect is assumed to lie in a finite linear span, cause and effect are identifiable from their observational distribution. This generalizes previous identifiability results based on Functional Causal Models with additive, heteroscedastic noise, etc. Since QPE resides entirely at the observational level, the parametric assumption does not require considering mechanisms, noise, or the Markov assumption, but rather directly utilizes the asymmetry of shape characteristics in the observational distribution. By performing basis function tests on the estimated QPE, causal directions can be distinguished. For multivariate causal discovery, Fisher Information is sufficient as

What carries the argument

Quantile Partial Effect (QPE) obtained from conditional quantile regression, which measures the shift in the outcome distribution at different probability levels and is assumed to lie in a finite linear span.

If this is right

  • Basis function tests on the estimated QPE distinguish causal directions in bivariate settings.
  • Fisher information recovers causal order in multivariate settings once the second moment of the QPE is bounded.
  • The method applies without specifying functional mechanisms or noise distributions.
  • The procedures succeed on a wide range of synthetic and real bivariate and multivariate causal discovery datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite-span assumption could be relaxed to other function classes such as smoothness or sparsity constraints to enlarge the set of identifiable models.
  • The link between QPE and score functions suggests possible integration with other information-based causal discovery techniques.
  • Similar quantile-effect tests might be developed for time-series or longitudinal data where the shape asymmetry evolves over time.

Load-bearing premise

The quantile partial effect of the cause on the effect must lie inside some finite-dimensional linear span.

What would settle it

A dataset with known true direction in which the QPE of cause on effect requires an infinite basis yet the finite-basis test recovers the correct direction, or a dataset in which the finite-span condition holds but the recovered direction is incorrect.

Figures

Figures reproduced from arXiv: 2509.12981 by Dehui Du, Xingzhe Sun, Yikang Chen.

Figure 1
Figure 1. Figure 1: Distributions and their QPEs for ANM Y = sin(X) + U and HNM Y = X3 + (1 + tanh((X − 1)2 ))U. (a) Joint distribution (heatmap) and samples (scatterplot); (b) Conditional density of Y |X (heatmap), conditional quantiles (white curves), and their gradients (white arrows); (c) QPE of Y |X (3D surface) and its intersection with the Y-Z plane (white curves); (d) Conditional density and conditional quantiles of X… view at source ↗
Figure 2
Figure 2. Figure 2: True and estimated QPEs of Y | X at samples from HNM Y = X3 + (1 + tanh((X − 1)2 ))U. From left to right: (a) True QPE; (b) QPE-k (Section 3.3); (c) Causal velocity model (Xi et al., 2025) (V-NN); (d) QPE-f (Section 3.4). The black lines represent the intersection of the QPE surface with the Y-Z plane. Only QPE-f’s trend along the Y-axis tend to match the true QPE. causal flow uθ, we can compute ∇xuθ and ∂… view at source ↗
Figure 3
Figure 3. Figure 3: ODRs of FICO and baselines on multivariate causal discovery datasets. Lower is better. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence behavior of SKEW, SCORE, and FICO on HNM-GP datasets. [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ODRs of FICO and baselines on 12 multivariate causal discovery datasets. Lower is better. [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relationship between FICO’s ODR and hyperparameters [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
read the original abstract

Quantile Partial Effect (QPE) is a statistic associated with conditional quantile regression, measuring the effect of covariates at different levels. Our theory demonstrates that when the QPE of cause on effect is assumed to lie in a finite linear span, cause and effect are identifiable from their observational distribution. This generalizes previous identifiability results based on Functional Causal Models (FCMs) with additive, heteroscedastic noise, etc. Meanwhile, since QPE resides entirely at the observational level, this parametric assumption does not require considering mechanisms, noise, or even the Markov assumption, but rather directly utilizes the asymmetry of shape characteristics in the observational distribution. By performing basis function tests on the estimated QPE, causal directions can be distinguished, which is empirically shown to be effective in experiments on a large number of bivariate causal discovery datasets. For multivariate causal discovery, leveraging the close connection between QPE and score functions, we find that Fisher Information is sufficient as a statistical measure to determine causal order when assumptions are made about the second moment of QPE. We validate the feasibility of using Fisher Information to identify causal order on multiple synthetic and real-world multivariate causal discovery datasets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using Quantile Partial Effect (QPE) for causal discovery. It demonstrates that assuming the QPE of the cause on the effect lies in a finite linear span allows identification of cause and effect from the observational distribution alone. This generalizes results from functional causal models with additive or heteroscedastic noise without invoking mechanisms, noise distributions, or the Markov assumption. For bivariate cases, basis function tests on estimated QPE distinguish directions, while for multivariate cases, Fisher information is used based on the connection between QPE and score functions under second-moment assumptions on QPE. The approach is empirically validated on multiple bivariate and multivariate synthetic and real-world datasets.

Significance. If the theoretical results hold, the paper contributes a new observational asymmetry for causal identifiability that operates directly at the level of quantile statistics, potentially extending causal discovery to settings where traditional assumptions do not apply. The empirical results on a large number of datasets provide evidence of practical utility. The connection to Fisher information offers a statistical measure for causal ordering in higher dimensions.

major comments (2)
  1. [§3, main identifiability theorem] The central identifiability claim requires that the finite linear span assumption on QPE(cause on effect) holds asymmetrically. The manuscript does not provide an explicit condition or proof ensuring that the reverse QPE(effect on cause) does not also lie in a finite linear span for the same joint distribution (e.g., when both are low-degree polynomials). Without this, the basis-function test cannot guarantee unique direction identification, which is load-bearing for the claim that directions are distinguishable from observational data alone.
  2. [§4.2, Fisher information derivation] The multivariate claim that Fisher Information suffices to determine causal order relies on assumptions about the second moment of QPE and its link to score functions. A detailed derivation is needed showing how the second-moment condition produces the ordering statistic, including regularity conditions and any error analysis.
minor comments (2)
  1. [Introduction] The definition and notation for QPE should be stated with an explicit equation in the introduction or preliminaries section for clarity.
  2. [Experiments] In the experiments section, specify the choice of basis functions, how the span dimension is selected, and any sensitivity analysis performed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the revisions we will make to improve the manuscript.

read point-by-point responses

  1. Referee: [§3, main identifiability theorem] The central identifiability claim requires that the finite linear span assumption on QPE(cause on effect) holds asymmetrically. The manuscript does not provide an explicit condition or proof ensuring that the reverse QPE(effect on cause) does not also lie in a finite linear span for the same joint distribution (e.g., when both are low-degree polynomials). Without this, the basis-function test cannot guarantee unique direction identification, which is load-bearing for the claim that directions are distinguishable from observational data alone.

    Authors: The identifiability result in Section 3 is conditional on the assumption that the QPE in the true causal direction lies in a finite linear span. The basis-function test then compares the fit of the estimated QPE to this span in each candidate direction and selects the direction with superior fit. We acknowledge that the manuscript does not explicitly characterize the set of distributions for which the assumption holds symmetrically. Such symmetric cases exist (e.g., linear or low-degree polynomial relationships), and in those cases the direction is not identifiable from the observational distribution alone, consistent with classical results. For generic nonlinear relationships the assumption is typically asymmetric. In the revision we will add a remark after Theorem 3 that (i) states the conditional nature of the result, (ii) gives a simple sufficient condition for asymmetry (nonlinearity of the conditional quantile function outside a finite-dimensional subspace), and (iii) notes that the test statistic remains well-defined and selects the better-fitting direction even when both directions satisfy the assumption to different degrees. revision: yes

  2. Referee: [§4.2, Fisher information derivation] The multivariate claim that Fisher Information suffices to determine causal order relies on assumptions about the second moment of QPE and its link to score functions. A detailed derivation is needed showing how the second-moment condition produces the ordering statistic, including regularity conditions and any error analysis.

    Authors: We agree that the current derivation in Section 4.2 is concise and would benefit from additional detail. In the revision we will expand the section to include: (1) a step-by-step derivation starting from the definition of the quantile partial effect, through its second-moment assumption, to the explicit expression for the Fisher information matrix as an ordering statistic; (2) the precise regularity conditions (twice continuous differentiability of the conditional density, finite second moments of the QPE, and integrability of the score); and (3) a short finite-sample error bound relating the estimation error of the QPE (via quantile regression) to the error in the estimated Fisher information. These additions will be placed immediately after the current statement of the multivariate procedure. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained without circular reductions

full rationale

The paper's central identifiability result rests on an explicit parametric assumption that the QPE of the true cause on the effect lies in a finite linear span, combined with basis-function tests applied to estimated QPE from the observational joint. This structure does not reduce any prediction or identifiability claim to a fitted quantity by construction, nor does it rely on self-citation load-bearing or imported uniqueness theorems. The multivariate extension invokes a connection to score functions and Fisher information under stated second-moment assumptions on QPE, but these remain observational statistics whose validity is tested empirically rather than forced by redefinition. No step equates the target causal order to an input fit or renames a known result; the derivation therefore retains independent content from the stated assumption and the proposed asymmetry tests.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach rests primarily on the domain assumption regarding the linear span of QPE and the second moment condition, with no new invented entities or free parameters explicitly fitted to the causal result itself.

free parameters (1)
  • linear span dimension for QPE
    The finite dimension of the linear span in which QPE is assumed to lie may require selection or fitting in application.
axioms (2)
  • domain assumption The quantile partial effect of the cause on the effect lies in a finite linear span
    This is the core parametric assumption enabling identifiability directly from the observational distribution.
  • domain assumption Assumptions about the second moment of QPE for using Fisher Information in multivariate settings
    Required for determining causal order in the multivariate case.

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