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Local Markov Equivalence for PC-style Local Causal Discovery and Identification of Controlled Direct Effects

Timothée Loranchet, Charles K. Assaad

TL;DR

This work describes a local class of graphs, defined relative to a target variable, that share a specific subset of $d-separations, and introduces a graphical representation of this class, called the local essential graph (LEG), and presents LocPC, a novel algorithm designed to recover the LEG from an observed distribution using only local conditional independence tests.

Abstract

Understanding and identifying controlled direct effects (CDEs) is crucial across numerous scientific domains, including public health. While existing methods can identify these effects from causal directed acyclic graphs (DAGs), the true underlying structure is often unknown in practice. Essential graphs, which represent a Markov equivalence class of DAGs characterized by the same set of $d$-separations, provide a more practical and realistic alternative. However, learning the full essential graph is computationally intensive and typically depends on strong, untestable assumptions. In this work, we characterize a local class of graphs, defined relative to a target variable, that share a specific subset of $d$-separations, and introduce a graphical representation of this class, called the local essential graph (LEG). We then present LocPC, a novel algorithm designed to recover the LEG from an observed distribution using only local conditional independence tests. Building on LocPC, we propose LocPC-CDE, an algorithm that discovers the portion of the LEG that is both sufficient and necessary to identify a CDE, bypassing the need of retrieving the full essential graph. Compared to global methods, our algorithms require less conditional independence tests and operate under weaker assumptions while maintaining theoretical guarantees. We illustrate the effectiveness of our approach through simulation studies.

Local Markov Equivalence for PC-style Local Causal Discovery and Identification of Controlled Direct Effects

TL;DR

This work describes a local class of graphs, defined relative to a target variable, that share a specific subset of $d-separations, and introduces a graphical representation of this class, called the local essential graph (LEG), and presents LocPC, a novel algorithm designed to recover the LEG from an observed distribution using only local conditional independence tests.

Abstract

Understanding and identifying controlled direct effects (CDEs) is crucial across numerous scientific domains, including public health. While existing methods can identify these effects from causal directed acyclic graphs (DAGs), the true underlying structure is often unknown in practice. Essential graphs, which represent a Markov equivalence class of DAGs characterized by the same set of -separations, provide a more practical and realistic alternative. However, learning the full essential graph is computationally intensive and typically depends on strong, untestable assumptions. In this work, we characterize a local class of graphs, defined relative to a target variable, that share a specific subset of -separations, and introduce a graphical representation of this class, called the local essential graph (LEG). We then present LocPC, a novel algorithm designed to recover the LEG from an observed distribution using only local conditional independence tests. Building on LocPC, we propose LocPC-CDE, an algorithm that discovers the portion of the LEG that is both sufficient and necessary to identify a CDE, bypassing the need of retrieving the full essential graph. Compared to global methods, our algorithms require less conditional independence tests and operate under weaker assumptions while maintaining theoretical guarantees. We illustrate the effectiveness of our approach through simulation studies.
Paper Structure (35 sections, 18 theorems, 25 equations, 5 figures, 2 algorithms)

This paper contains 35 sections, 18 theorems, 25 equations, 5 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related Work
  3. Preliminaries
  4. Characterization of Local Markov Equivalence
  5. Local causal discovery
  6. Local causal discovery for identifying CDE
  7. Experiments
  8. Conclusion
  9. About spurious neighbors and descendant inducing paths
  10. A simple example of spurious neighbor
  11. Illustration of descendant inducing paths
  12. Why not all descendant inducing neighbors are spurious neighbors?
  13. An example of a LEG containing spurious neighbors
  14. Proofs
  15. Proof of Theorem \ref{['prop:spurious']}
  16. ...and 20 more sections

Key Result

Theorem 1

Let $\mathcal{G} := (\mathbb V, \mathbb E)$ be a DAG and let $D \in \mathbb V$. We have $SNe(D,\mathcal{G})\subseteq DINe(D,\mathcal{G})\subset De(D,\mathcal{G}).$

Figures (5)

  • Figure 1: A DAG $\mathcal{G}$ and the LEGs $\mathcal{L}^{Y,0}$, $\mathcal{L}^{Y,1}$, and $\mathcal{L}^{Y,2}$ around node $Y$. Red: outcome/target $Y$; blue: treatment $X$; grey: $h$-neighborhood nodes; red arrow: direct effect ($M$: mediator).
  • Figure 2: DAG $\mathcal{G}$, essential graph $\mathcal{C}$, and LEG $\mathcal{L}^{Y,1}$. $\mathbb D = \{Y, X, D_1\}$ satisfies the NOC (Def. \ref{['def:orientability']}), so Theorem \ref{['corollary:identifiability_CDE']} implies that $CDE(X \textcolor{red}{\to} Y)$ is not identifiable, even with global discovery.
  • Figure 3: Empirical performance of LocPC-CDE across different graph sizes and SCM settings, compared to global discovery (PC) and state-of-the-art local discovery methods.
  • Figure 4: A DAG $\mathcal{G}$ and the LEGs $\mathcal{L}^{Y,0}$, $\mathcal{L}^{Y,1}$, and $\mathcal{L}^{Y,2}$ around node $Y$. Red: outcome/target $Y$; blue: treatment $X$; grey: $h$-neighborhood nodes; red arrow: direct effect ($M$: mediator).
  • Figure 5: Empirical number of CI tests performed by LocPC as a function of discovered hop $h$ (left), sample size $n$ (middle), and edge probability $p$ (right).

Theorems & Definitions (34)

  • Definition 1: Spurious neighbors
  • Definition 2: Descendant Inducing Path, DIP
  • Theorem 1: Spurious are descendant inducing neighbors
  • Definition 3: Local Markov equivalence class, LMEC
  • Theorem 2: Structural characteristics of LMEC's DAGs
  • Definition 4: Local essential graph, LEG
  • Theorem 3
  • Corollary 1
  • Definition 5: CI-based NNC rule
  • Theorem 4: Correctness of LocPC
  • ...and 24 more