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Identifying Causal Effects via Context-specific Independence Relations

Santtu Tikka, Antti Hyttinen, Juha Karvanen

TL;DR

The paper addresses identifying causal effects when context-specific independence (CSI) relations hold, showing that standard do-calculus may be incomplete in such settings. It introduces LDAGs to encode CSIs and develops a CSI-calculus that extends do-calculus, along with an automated forward search (Algorithm 1) that derives identifiability formulas under CSIs using separation criteria and context pruning. The authors prove NP-hardness of deciding non-identifiability with CSIs and demonstrate that the CSI-calculus can yield identifying formulas beyond do-calculus, enabling identifiability in cases previously deemed non-identifiable. The work lays a foundation for scalable CSI-enabled identifiability and has potential implications for related problems like transportability and missing data, while also outlining directions for extending to broader variable types and completeness results.

Abstract

Causal effect identification considers whether an interventional probability distribution can be uniquely determined from a passively observed distribution in a given causal structure. If the generating system induces context-specific independence (CSI) relations, the existing identification procedures and criteria based on do-calculus are inherently incomplete. We show that deciding causal effect non-identifiability is NP-hard in the presence of CSIs. Motivated by this, we design a calculus and an automated search procedure for identifying causal effects in the presence of CSIs. The approach is provably sound and it includes standard do-calculus as a special case. With the approach we can obtain identifying formulas that were unobtainable previously, and demonstrate that a small number of CSI-relations may be sufficient to turn a previously non-identifiable instance to identifiable.

Identifying Causal Effects via Context-specific Independence Relations

TL;DR

The paper addresses identifying causal effects when context-specific independence (CSI) relations hold, showing that standard do-calculus may be incomplete in such settings. It introduces LDAGs to encode CSIs and develops a CSI-calculus that extends do-calculus, along with an automated forward search (Algorithm 1) that derives identifiability formulas under CSIs using separation criteria and context pruning. The authors prove NP-hardness of deciding non-identifiability with CSIs and demonstrate that the CSI-calculus can yield identifying formulas beyond do-calculus, enabling identifiability in cases previously deemed non-identifiable. The work lays a foundation for scalable CSI-enabled identifiability and has potential implications for related problems like transportability and missing data, while also outlining directions for extending to broader variable types and completeness results.

Abstract

Causal effect identification considers whether an interventional probability distribution can be uniquely determined from a passively observed distribution in a given causal structure. If the generating system induces context-specific independence (CSI) relations, the existing identification procedures and criteria based on do-calculus are inherently incomplete. We show that deciding causal effect non-identifiability is NP-hard in the presence of CSIs. Motivated by this, we design a calculus and an automated search procedure for identifying causal effects in the presence of CSIs. The approach is provably sound and it includes standard do-calculus as a special case. With the approach we can obtain identifying formulas that were unobtainable previously, and demonstrate that a small number of CSI-relations may be sufficient to turn a previously non-identifiable instance to identifiable.

Paper Structure

This paper contains 25 sections, 6 theorems, 13 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries: Graphical Models for Context-specific Independence
  3. Causal Effect Identification for CSI-based Graphical Models
  4. On Computational Complexity
  5. A Calculus for Determining Identifiability
  6. Rule 1
  7. Rule 2
  8. Rule 3
  9. Reduction to the Identifiability of Conditional Probabilities in LDAGs
  10. Rules of the Calculus
  11. Rule 1
  12. Rule 2
  13. Rule 3
  14. Rule 4
  15. Rule 5
  16. ...and 10 more sections

Key Result

Theorem 1

Deciding non-identifiability of a causal effect given an LDAG over $\boldsymbol V$ and a passively observed distribution over $\boldsymbol W \subseteq \boldsymbol V$ is NP-hard.

Figures (11)

  • Figure 1: (a) L is latent unobserved variable. (b) CPT for $P(X { \, | \, } A,L)$. (c) Decision tree with $P(X { \, | \, } A,L)$ given in the leaf nodes. (d) corresponding labeled DAG (LDAG). (e) LDAG with an intervention node added for $X$.
  • Figure 2: Rules of do-calculus. The sets $\boldsymbol X,\boldsymbol Y,\boldsymbol Z$ and $\boldsymbol W$ are disjoint. Notation ${ \, || \, } \boldsymbol X$ means that the condition is evaluated in a graph in which edges into $\boldsymbol X$ are removed. $\boldsymbol I_{\boldsymbol Z}$ denotes the intervention nodes of variables $\boldsymbol Z$ (see Sec. \ref{['sec:reduction']}).
  • Figure 3: Rules of CSI-calculus. The sets $\boldsymbol X_1,\boldsymbol X_2,\boldsymbol Y_1, \boldsymbol Y_2, \boldsymbol Z_1$ and $\boldsymbol Z_2$ are disjoint. We write $\boldsymbol w$ as shorthand for the explicit assignment $\boldsymbol W = \boldsymbol w$.
  • Figure 4: A derivation of $P(Y { \, | \, } \textrm{do}(X))$ from $P(X,Y,A)$ in the example of Fig. \ref{['fig:intro_example']}. The applied rules and CSIs are marked next to the edges connecting the terms. The identifying formula is Eq. \ref{['eq:running']}.
  • Figure 5:
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: Soundness
  • Theorem 6