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Meta-Dependence in Conditional Independence Testing

Bijan Mazaheri, Jiaqi Zhang, Caroline Uhler

TL;DR

This work tackles the problem of meta-dependence among conditional independence tests used in constraint-based causal discovery by introducing CIMD, an information-theoretic measure based on information projections. CIMD quantifies how projecting a distribution onto one CI constraint changes its proximity to another CI constraint, capturing shared information between CI tests and the resulting co-occurrence of independence judgments. The authors provide practical computation methods, including an MLE-based approach for general distributions and closed-form formulas for multivariate Gaussians, and validate the approach with synthetic and real-world datasets, showing alignment between CIMD and finite-sample CI co-occurrences (FS-CID). The results offer a principled way to adjust significance thresholds in causal discovery and point toward robust “confidence ensembles” of causal structures under distributional uncertainty, with broad applicability beyond Gaussian models.

Abstract

Constraint-based causal discovery algorithms utilize many statistical tests for conditional independence to uncover networks of causal dependencies. These approaches to causal discovery rely on an assumed correspondence between the graphical properties of a causal structure and the conditional independence properties of observed variables, known as the causal Markov condition and faithfulness. Finite data yields an empirical distribution that is "close" to the actual distribution. Across these many possible empirical distributions, the correspondence to the graphical properties can break down for different conditional independencies, and multiple violations can occur at the same time. We study this "meta-dependence" between conditional independence properties using the following geometric intuition: each conditional independence property constrains the space of possible joint distributions to a manifold. The "meta-dependence" between conditional independences is informed by the position of these manifolds relative to the true probability distribution. We provide a simple-to-compute measure of this meta-dependence using information projections and consolidate our findings empirically using both synthetic and real-world data.

Meta-Dependence in Conditional Independence Testing

TL;DR

This work tackles the problem of meta-dependence among conditional independence tests used in constraint-based causal discovery by introducing CIMD, an information-theoretic measure based on information projections. CIMD quantifies how projecting a distribution onto one CI constraint changes its proximity to another CI constraint, capturing shared information between CI tests and the resulting co-occurrence of independence judgments. The authors provide practical computation methods, including an MLE-based approach for general distributions and closed-form formulas for multivariate Gaussians, and validate the approach with synthetic and real-world datasets, showing alignment between CIMD and finite-sample CI co-occurrences (FS-CID). The results offer a principled way to adjust significance thresholds in causal discovery and point toward robust “confidence ensembles” of causal structures under distributional uncertainty, with broad applicability beyond Gaussian models.

Abstract

Constraint-based causal discovery algorithms utilize many statistical tests for conditional independence to uncover networks of causal dependencies. These approaches to causal discovery rely on an assumed correspondence between the graphical properties of a causal structure and the conditional independence properties of observed variables, known as the causal Markov condition and faithfulness. Finite data yields an empirical distribution that is "close" to the actual distribution. Across these many possible empirical distributions, the correspondence to the graphical properties can break down for different conditional independencies, and multiple violations can occur at the same time. We study this "meta-dependence" between conditional independence properties using the following geometric intuition: each conditional independence property constrains the space of possible joint distributions to a manifold. The "meta-dependence" between conditional independences is informed by the position of these manifolds relative to the true probability distribution. We provide a simple-to-compute measure of this meta-dependence using information projections and consolidate our findings empirically using both synthetic and real-world data.

Paper Structure

This paper contains 39 sections, 2 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Significance Thresholds.
  4. Validating Causal Discovery.
  5. Uncertainty in Causal Discovery.
  6. Contributions
  7. Related Works
  8. Causal Strength and Stability.
  9. Complexity of DAGs.
  10. Separating Set Size.
  11. Falsifying Causal Models.
  12. Preliminaries
  13. Notation.
  14. Structural Causal Models.
  15. Faithfulness.
  16. ...and 24 more sections

Key Result

Lemma 6.1

Fix some function-class for the structural equations and let $\mathbb{P}_{\mathcal{G}}$ be the set of distributions generated according to the DAG $\mathcal{G}$. Projecting $\hat{P}$ into $\mathbb{P}_{\mathcal{G}}$ corresponds to composing maximum likelihood estimators in the factorization of $\math where $\hat{P}(v \:\vert\: \operatorname{\mathbf{pa}}^{\mathbf{v}}(v))$ are the MLEs for predicting

Figures (4)

  • Figure 1: (a) Dependence of faithfulness for structural equations given in Eq. \ref{['eq: neg dependence']} when $\alpha_1 = .5$. The shaded region between the red and blue lines corresponds to models where moving towards blue moves away from red. Hence, the shaded region has a "negative" dependence between faithfulness violations $A \mathrel{\hbox{$\perp$}\mkern2mu{\perp}} C$ and $B \mathrel{\hbox{$\perp$}\mkern2mu{\perp}} C$, while the unshaded region has a "positive" dependence. (b) The introduced metric, CIMD (Section \ref{['sec: definition']}), measuring the meta-dependence between CI-tests, computed from the covariance matrix. (c) Estimation of CIMD with finite samples (Sections \ref{['sec: empirica']} and \ref{['sec: empirical']}).
  • Figure 2: Meta-dependencies of CI-tests as measured by FS-CID in (a) and CMID-lim in (b).
  • Figure 3: Both (a) and (b) plot the CIMD for various parameters from the structural equations given in Eq. \ref{['eq: neg dependence']}. The effective structure in (a) is a Markov chain, and the effective structure in (b) is a collider.
  • Figure 4: Demonstrating the dependence between CI-tests in real-world data using both FS-CID and limited CIMD.

Theorems & Definitions (5)

  • Definition 3.1: $\lambda$-strong faithfulness
  • Definition 5.1: CIMD
  • Lemma 6.1
  • proof
  • Theorem 6.2