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Causal Discovery via Quantile Partial Effect

Yikang Chen, Xingzhe Sun, Dehui Du

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.

Causal Discovery via Quantile Partial Effect

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.

Paper Structure

This paper contains 61 sections, 16 theorems, 54 equations, 7 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and General Assumptions
  4. Structural Causal Models
  5. Score Function and Stein's Identity
  6. Causal Normalizing Flows
  7. QPE for Causal Discovery
  8. Definition of QPE and Equivalent Concepts
  9. Cause-Effect Identifiability by QPE in Finite Linear Span
  10. Proof
  11. QPE for Bivariate Causal Discovery
  12. Quantify Cause-Effect Identifiability through the Lens of FCM
  13. Kernel-based QPE with Least Square Basis Test
  14. Kernel-based QPE
  15. Least Square Basis Test
  16. ...and 46 more sections

Key Result

Proposition 3.1

$\bm{\psi}_{Y\mid\bm{X}}=-\nabla_{\bm{x}}F_{Y\mid\bm{X}}/\partial_yF_{Y\mid\bm{X}}=-\nabla_{\bm{x}}F_{Y\mid\bm{X}}/p_{Y\mid\bm{X}}$.

Figures (7)

  • Figure 1: Distributions and their QPEs for ANM $Y=\sin(X)+U$ and HNM $Y=X^3+(1+\tanh((X-1)^2))\,U$. (a) Joint distribution (heatmap) and samples (scatterplot); (b) Conditional density of $Y\!\mid\!X$ (heatmap), conditional quantiles (white curves), and their gradients (white arrows); (c) QPE of $Y\!\mid\!X$ (3D surface) and its intersection with the Y-Z plane (white curves); (d) Conditional density and conditional quantiles of $X\!\mid\!Y$; (e) QPE of $X\!\mid\!Y$ (3D surface) and its intersection with the X-Z plane. ANM guarantees that the intersection of the QPE of $Y\!\mid\!X$ in the Y-Z plane is a constant function, while HNM guarantees it is an affine function, due to restrictions on the QPE form (see \ref{['tab:fcm-qpe-basis']}); the converse generally does not hold (\ref{['ssec:identify']}).
  • Figure 2: For clarity, we present a dependency graph of the assumptions and theorems in the main text, where "A$\rightarrow$B" indicates that B depends on A: (a) Theorems in correspond to \ref{['sec:qpe_cd']}; (b) Theorems in correspond to \ref{['sec:qpe_bi_cd']}; (c) Theorems in correspond to \ref{['sec:fi_mul_cd']}.
  • Figure 3: True and estimated QPEs of $Y\!\mid\!X$ at samples from HNM $Y=X^3+(1+\tanh((X-1)^2))\,U$. From left to right: (a) True QPE; (b) QPE-k (\ref{['ssec:qpe-k']}); (c) Causal velocity model Xi2025 (V-NN); (d) QPE-f (\ref{['ssec:qpe-f']}). The black lines represent the intersection of the QPE surface with the Y-Z plane. Only QPE-f's trend tends to match the true QPE in high-density areas.
  • Figure 4: Convergence behavior of SKEW, SCORE, and FICO on HNM-GP datasets.
  • Figure 5: Relationship between FICO's ODR and hyperparameters $\alpha$ and $\beta$ under the heteroscedastic Gaussian assumption. (a) 10-variable ER graph; (b) 20-variable ER graph. The expected numbers of edges in these graphs is 4 times their dimensions.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 3.1: Quantile Partial Effect
  • Proposition 3.1: QPE from CDF
  • Proposition 3.1: Causal Velocity is QPE
  • Lemma 3.1
  • Theorem 3.3: Identifiability of QPE in Finite Linear Span
  • Corollary 3.4: Cause-Effect Identifiability by QPE in Finite Linear Span
  • Corollary 4.1
  • Theorem 5.2
  • Corollary 5.2
  • Proposition A.0: QPE from CDF
  • ...and 19 more