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Causal Additive Models with Unobserved Causal Paths and Backdoor Paths

Thong Pham, Takashi Nicholas Maeda, Shohei Shimizu

TL;DR

This work derives conditions that enable identification of the parent-child relationship in a bow, an adjacent pair of observed variables sharing a hidden common parent, and provides a sound and complete algorithm that incorporates these insights.

Abstract

Causal additive models provide a tractable yet expressive framework for causal discovery in the presence of hidden variables. However, when unobserved backdoor or causal paths exist between two variables, their causal relationship is often unidentifiable under existing theories. We establish sufficient conditions under which causal directions can be identified in many such cases. In particular, we derive conditions that enable identification of the parent-child relationship in a bow, an adjacent pair of observed variables sharing a hidden common parent. This represents a notoriously difficult case in causal discovery, and, to our knowledge, no prior work has established such identifiability in any causal model without imposing assumptions on the hidden variables. Our conditions rely on new characterizations of regression sets and a hybrid approach that combines independence among regression residuals with conditional independencies among observed variables. We further provide a sound and complete algorithm that incorporates these insights, and empirical evaluations demonstrate competitive performance with state-of-the-art methods.

Causal Additive Models with Unobserved Causal Paths and Backdoor Paths

TL;DR

This work derives conditions that enable identification of the parent-child relationship in a bow, an adjacent pair of observed variables sharing a hidden common parent, and provides a sound and complete algorithm that incorporates these insights.

Abstract

Causal additive models provide a tractable yet expressive framework for causal discovery in the presence of hidden variables. However, when unobserved backdoor or causal paths exist between two variables, their causal relationship is often unidentifiable under existing theories. We establish sufficient conditions under which causal directions can be identified in many such cases. In particular, we derive conditions that enable identification of the parent-child relationship in a bow, an adjacent pair of observed variables sharing a hidden common parent. This represents a notoriously difficult case in causal discovery, and, to our knowledge, no prior work has established such identifiability in any causal model without imposing assumptions on the hidden variables. Our conditions rely on new characterizations of regression sets and a hybrid approach that combines independence among regression residuals with conditional independencies among observed variables. We further provide a sound and complete algorithm that incorporates these insights, and empirical evaluations demonstrate competitive performance with state-of-the-art methods.

Paper Structure

This paper contains 50 sections, 18 theorems, 26 equations, 7 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The causal model
  4. Unobserved backdoor paths and unobserved causal paths
  5. Visible and invisible pairs
  6. Identifying causal relationships in invisible pairs
  7. Utilizing independence between regression residuals
  8. Utilizing Conditional Independence
  9. Search methods
  10. Limitations of the CAM-UV algorithm
  11. Proposed search method
  12. Experiments
  13. Illustrative examples
  14. Random graph experiment
  15. Real-world sociology data
  16. ...and 35 more sections

Key Result

Lemma 1

Consider $X' \subseteq X$ and $x_i,x_j \in X'$. $x_j$ is a parent of $x_i$ and there is no UBP or UCP between $x_j$ and $x_i$ with respect to $X'$ if and only if In this case, we refer to $x_j$ as a visible parent of $x_i$ with respect to $X'$. When omitted, $X'$ is taken to be $X$.

Figures (7)

  • Figure 1: Examples of identifying causal relationships in the presence of UBPs/UCPs.
  • Figure 2: Performance on the graph in Figs. \ref{['fig:illustrative_1']}a and b.
  • Figure 3: Performance in BA random graphs with Gaussian noises. a: identifying the adjacency matrix, b: identifying ancestorships.
  • Figure 4: Results on sociology data. a: ground truth based on domain knowledge Duncan72book. A bidirected edge indicates that the relation is not modeled. A dashed directed edge represents an ancestor relationship. b: result by CAM-UV. A solid directed edge denotes a visible parent–child relationship (adjacency). An empty edge denotes a visible non-edge (non-adjacency). A bidirected edge denotes an invisible pair. c: result of CAM-UV-X. Solid directed/bidirected edges and empty edges have the same meaning as those of CAM-UV. A dashed undirected edge in blue, unique to CAM-UV-X and appearing only in an invisible pair connected by a bidirected edge, indicates non-adjacency.
  • Figure B.1: Examples where the regression approach alone is sufficient to resolve parent-child relationships in an invisible pair.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 2.1: Unobserved Causal Path
  • Definition 2.2: Unobserved Backdoor Path
  • Lemma 1: visible parent
  • Lemma 2: visible non-edge
  • Lemma 3: invisible pairs
  • Lemma 4
  • Lemma 5
  • Remark 1
  • Lemma 6
  • Lemma 7
  • ...and 19 more