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Unfaithful Probability Distributions in Binary Triple of Causality Directed Acyclic Graph

Jingwei Liu

TL;DR

This paper addresses faithfulness in causal DAGs for binary triples $(X,Y,Z)$ by constructing explicit unfaithful probability distributions and a general family of unfaithful distributions within the triple DAG framework. It develops the Markov-factorization and independence concepts, proves Theorems 1–2 that connect independence patterns to faithful graphs, and provides a suite of explicit unfaithful examples that illustrate how multiple independences can co-occur without a faithful graphical representation. The core result is Theorem 3, which organizes a broad class of unfaithful distributions in the triple DAG and shows that, except for a special case where $\mathscr{I}_{X:1}(P)=\mathscr{I}_{\bigcap}(P)$, most configurations are unfaithful to existing Fig. 1/Fig. 2 DAGs, highlighting potential PC algorithm failures. The findings have practical implications for causal discovery and motivate refinements to PC-based methods within the Spirtes–Glymour–Scheines framework, especially when multiple independence structures arise simultaneously.

Abstract

Faithfulness is the foundation of probability distribution and graph in causal discovery and causal inference. In this paper, several unfaithful probability distribution examples are constructed in three--vertices binary causality directed acyclic graph (DAG) structure, which are not faithful to causal DAGs described in J.M.,Robins,et al. Uniform consistency in causal inference. Biometrika (2003),90(3): 491--515. And the general unfaithful probability distribution with multiple independence and conditional independence in binary triple causal DAG is given.

Unfaithful Probability Distributions in Binary Triple of Causality Directed Acyclic Graph

TL;DR

This paper addresses faithfulness in causal DAGs for binary triples by constructing explicit unfaithful probability distributions and a general family of unfaithful distributions within the triple DAG framework. It develops the Markov-factorization and independence concepts, proves Theorems 1–2 that connect independence patterns to faithful graphs, and provides a suite of explicit unfaithful examples that illustrate how multiple independences can co-occur without a faithful graphical representation. The core result is Theorem 3, which organizes a broad class of unfaithful distributions in the triple DAG and shows that, except for a special case where , most configurations are unfaithful to existing Fig. 1/Fig. 2 DAGs, highlighting potential PC algorithm failures. The findings have practical implications for causal discovery and motivate refinements to PC-based methods within the Spirtes–Glymour–Scheines framework, especially when multiple independence structures arise simultaneously.

Abstract

Faithfulness is the foundation of probability distribution and graph in causal discovery and causal inference. In this paper, several unfaithful probability distribution examples are constructed in three--vertices binary causality directed acyclic graph (DAG) structure, which are not faithful to causal DAGs described in J.M.,Robins,et al. Uniform consistency in causal inference. Biometrika (2003),90(3): 491--515. And the general unfaithful probability distribution with multiple independence and conditional independence in binary triple causal DAG is given.

Paper Structure

This paper contains 7 sections, 5 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The binary three vertices causal DAG
  3. Unfaithful probability distribution in binary three vertices DAG
  4. General unfaithful probability distribution in Triple causal DAG
  5. Discussion
  6. Author contributions statement
  7. Acknowledgments

Key Result

Lemma 1

For random variables $(X,Y,Z)$, one has (1) (Symmetry) $X\perp\!\!\!\perp Y \Rightarrow Y\perp\!\!\!\perp X$, $X\perp\!\!\!\perp Y |Z \Rightarrow Y\perp\!\!\!\perp X|Z$. (2) (Decomposition) $X\perp\!\!\!\perp (Y,Z)$$\Rightarrow$ ($X\perp\!\!\!\perp Y$) and ($X\perp\!\!\!\perp Z$). (3) (Weak union)

Figures (2)

  • Figure 1: 3-vertices causal DAGs from $X$ to $Y$ with confounder $Z$.
  • Figure 2: 3-vertices causal DAGs from $X$ to $Y$, and $Z$ is not a confounder.

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • Theorem 1: Theorem 1
  • Example 1
  • Theorem 2: Theorem 2
  • proof
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • ...and 7 more