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On the Representation of Pairwise Causal Background Knowledge and Its Applications in Causal Inference

Zhuangyan Fang, Ruiqi Zhao, Yue Liu, Yangbo He

TL;DR

This work develops a unified, scalable framework for representing and exploiting pairwise causal background knowledge in causal inference. It introduces direct causal clauses (DCCs) to uniformly encode direct, ancestral, and non-ancestral constraints and proves that any knowledge set can be decomposed into a causal MPDAG plus a minimal residual DCC set. The decomposed MPDAG fully governs identifiability of causal effects, while residual DCCs refine the possible effect values when unidentified; the authors provide polynomial-time algorithms for consistency/equivalence checks and MPDAG construction, and extend IDA-type methods to estimate possible causal effects with background knowledge. Empirical results show that incorporating background knowledge, especially ancestral constraints, substantially improves identifiability, highlighting the practical value for observational causal analysis. The framework lays a foundation for robust causal discovery and inference under informative background knowledge, with future work on scalability, non-pairwise constraints, and hidden-variable settings.

Abstract

Pairwise causal background knowledge about the existence or absence of causal edges and paths is frequently encountered in observational studies. Such constraints allow the shared directed and undirected edges in the constrained subclass of Markov equivalent DAGs to be represented as a causal maximally partially directed acyclic graph (MPDAG). In this paper, we first provide a sound and complete graphical characterization of causal MPDAGs and introduce a minimal representation of a causal MPDAG. Then, we give a unified representation for three types of pairwise causal background knowledge, including direct, ancestral and non-ancestral causal knowledge, by introducing a novel concept called direct causal clause (DCC). Using DCCs, we study the consistency and equivalence of pairwise causal background knowledge and show that any pairwise causal background knowledge set can be uniquely and equivalently decomposed into the causal MPDAG representing the refined Markov equivalence class and a minimal residual set of DCCs. Polynomial-time algorithms are also provided for checking consistency and equivalence, as well as for finding the decomposed MPDAG and the residual DCCs. Finally, with pairwise causal background knowledge, we prove a sufficient and necessary condition to identify causal effects and surprisingly find that the identifiability of causal effects only depends on the decomposed MPDAG. We also develop a local IDA-type algorithm to estimate the possible values of an unidentifiable effect. Simulations suggest that pairwise causal background knowledge can significantly improve the identifiability of causal effects.

On the Representation of Pairwise Causal Background Knowledge and Its Applications in Causal Inference

TL;DR

This work develops a unified, scalable framework for representing and exploiting pairwise causal background knowledge in causal inference. It introduces direct causal clauses (DCCs) to uniformly encode direct, ancestral, and non-ancestral constraints and proves that any knowledge set can be decomposed into a causal MPDAG plus a minimal residual DCC set. The decomposed MPDAG fully governs identifiability of causal effects, while residual DCCs refine the possible effect values when unidentified; the authors provide polynomial-time algorithms for consistency/equivalence checks and MPDAG construction, and extend IDA-type methods to estimate possible causal effects with background knowledge. Empirical results show that incorporating background knowledge, especially ancestral constraints, substantially improves identifiability, highlighting the practical value for observational causal analysis. The framework lays a foundation for robust causal discovery and inference under informative background knowledge, with future work on scalability, non-pairwise constraints, and hidden-variable settings.

Abstract

Pairwise causal background knowledge about the existence or absence of causal edges and paths is frequently encountered in observational studies. Such constraints allow the shared directed and undirected edges in the constrained subclass of Markov equivalent DAGs to be represented as a causal maximally partially directed acyclic graph (MPDAG). In this paper, we first provide a sound and complete graphical characterization of causal MPDAGs and introduce a minimal representation of a causal MPDAG. Then, we give a unified representation for three types of pairwise causal background knowledge, including direct, ancestral and non-ancestral causal knowledge, by introducing a novel concept called direct causal clause (DCC). Using DCCs, we study the consistency and equivalence of pairwise causal background knowledge and show that any pairwise causal background knowledge set can be uniquely and equivalently decomposed into the causal MPDAG representing the refined Markov equivalence class and a minimal residual set of DCCs. Polynomial-time algorithms are also provided for checking consistency and equivalence, as well as for finding the decomposed MPDAG and the residual DCCs. Finally, with pairwise causal background knowledge, we prove a sufficient and necessary condition to identify causal effects and surprisingly find that the identifiability of causal effects only depends on the decomposed MPDAG. We also develop a local IDA-type algorithm to estimate the possible values of an unidentifiable effect. Simulations suggest that pairwise causal background knowledge can significantly improve the identifiability of causal effects.
Paper Structure (48 sections, 48 theorems, 24 equations, 18 figures, 7 algorithms)

This paper contains 48 sections, 48 theorems, 24 equations, 18 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Causal Graphical Models
  3. Pairwise Causal Background Knowledge
  4. Intervention Calculus
  5. A Graphical Characterization of Causal MPDAGs
  6. A Unified Representation of Pairwise Causal Constraints
  7. Direct Causal Clauses
  8. Equivalent Decomposition of Pairwise Causal Constraints
  9. Non-Pairwise Causal Background Knowledge
  10. Polynomial-Time Algorithms
  11. Algorithms for Checking Consistency and Equivalence
  12. Algorithms for Finding MPDAGs and Minimal DCC Sets
  13. Causal Inference with Background Knowledge
  14. Identifiability
  15. Estimating Possible Causal Effects
  16. ...and 33 more sections

Key Result

Theorem 1

A partially directed graph $\mathcal{H} = (\mathbf{V}, \mathbf{E})$ is a causal MPDAG if and only if $\mathcal{H}$ satisfies the following conditions.

Figures (18)

  • Figure 1: A CPDAG over three variables including smoking, bronchitis and dyspnea is given in Figure \ref{['fig:intro1']}, which represents the Markov equivalent class shown in Figures \ref{['fig:intro2']}.
  • Figure 2: A visualization of Meek's rules. If the graph on the left-hand side of a rule is an induced subgraph of a PDAG, then orient the undirected edge such that the resulting subgraph is the one on the right-hand side of the rule.
  • Figure 3: An example of a causal MPDAG. Let ${\cal B}=\{ X\dashrightarrow Y\}$ and $\cal H$ be the MPDAG of $[{\cal G}^*,\cal B]$. $[{\cal G}^*,\cal B]$ consists of the DAGs from ${\cal G}_1$ to ${\cal G}_4$, while $[{\cal H}]$ consists of the DAGs from ${\cal G}_1$ to ${\cal G}_6$, which indicates that $[{\cal G}^*,{\cal B}]\subsetneq[\cal H]$.
  • Figure 4: Examples to illustrate the graphical characterization of causal MPDAGs.
  • Figure 5: The five configurations of M-strongly protected edges.
  • ...and 13 more figures

Theorems & Definitions (123)

  • Definition 1: Completed PDAG, andersson1997characterization
  • Definition 2: Pairwise Causal Constraints
  • Definition 3: Restricted Markov Equivalence Class
  • Definition 4: Causal MPDAG
  • Example 1
  • Definition 5: Fully Informative MPDAG
  • Definition 6: B-component
  • Definition 7: Chain Skeleton
  • Theorem 1
  • Proposition 1
  • ...and 113 more