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When Selection Meets Intervention: Additional Complexities in Causal Discovery

Haoyue Dai, Ignavier Ng, Jianle Sun, Zeyu Tang, Gongxu Luo, Xinshuai Dong, Peter Spirtes, Kun Zhang

TL;DR

The paper tackles selection bias in interventional causal discovery by introducing an interventional twin graph that jointly models the observed world and the counterfactual, pre-intervention world. It establishes Markov properties and MAG-based equivalence criteria for data generated under soft interventions with unknown targets, and develops the CDIS algorithm to identify causal relations and selection mechanisms up to an equivalence class. Through simulations and real-world data (biology and education), it demonstrates improved identification of true causal relations despite selection, outperforming baselines in precision and robustness. This framework enables more reliable causal discovery in settings where enrollment is conditioned on selection, with practical impact for biomedical experiments and policy evaluations.

Abstract

We address the common yet often-overlooked selection bias in interventional studies, where subjects are selectively enrolled into experiments. For instance, participants in a drug trial are usually patients of the relevant disease; A/B tests on mobile applications target existing users only, and gene perturbation studies typically focus on specific cell types, such as cancer cells. Ignoring this bias leads to incorrect causal discovery results. Even when recognized, the existing paradigm for interventional causal discovery still fails to address it. This is because subtle differences in when and where interventions happen can lead to significantly different statistical patterns. We capture this dynamic by introducing a graphical model that explicitly accounts for both the observed world (where interventions are applied) and the counterfactual world (where selection occurs while interventions have not been applied). We characterize the Markov property of the model, and propose a provably sound algorithm to identify causal relations as well as selection mechanisms up to the equivalence class, from data with soft interventions and unknown targets. Through synthetic and real-world experiments, we demonstrate that our algorithm effectively identifies true causal relations despite the presence of selection bias.

When Selection Meets Intervention: Additional Complexities in Causal Discovery

TL;DR

The paper tackles selection bias in interventional causal discovery by introducing an interventional twin graph that jointly models the observed world and the counterfactual, pre-intervention world. It establishes Markov properties and MAG-based equivalence criteria for data generated under soft interventions with unknown targets, and develops the CDIS algorithm to identify causal relations and selection mechanisms up to an equivalence class. Through simulations and real-world data (biology and education), it demonstrates improved identification of true causal relations despite selection, outperforming baselines in precision and robustness. This framework enables more reliable causal discovery in settings where enrollment is conditioned on selection, with practical impact for biomedical experiments and policy evaluations.

Abstract

We address the common yet often-overlooked selection bias in interventional studies, where subjects are selectively enrolled into experiments. For instance, participants in a drug trial are usually patients of the relevant disease; A/B tests on mobile applications target existing users only, and gene perturbation studies typically focus on specific cell types, such as cancer cells. Ignoring this bias leads to incorrect causal discovery results. Even when recognized, the existing paradigm for interventional causal discovery still fails to address it. This is because subtle differences in when and where interventions happen can lead to significantly different statistical patterns. We capture this dynamic by introducing a graphical model that explicitly accounts for both the observed world (where interventions are applied) and the counterfactual world (where selection occurs while interventions have not been applied). We characterize the Markov property of the model, and propose a provably sound algorithm to identify causal relations as well as selection mechanisms up to the equivalence class, from data with soft interventions and unknown targets. Through synthetic and real-world experiments, we demonstrate that our algorithm effectively identifies true causal relations despite the presence of selection bias.

Paper Structure

This paper contains 32 sections, 14 theorems, 3 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Revisiting the Established Paradigm of Interventional Causal Discovery
  4. How the Augmented DAG Paradigm Fails in the Presence of Selection
  5. Why the Augmented DAG Paradigm Fails when Selection Presents
  6. Causal Model Involving Selection and Intervention
  7. Causal Graphical Model for Interventions Under Selection
  8. The Markov Properties
  9. Markov Equivalence Relations
  10. MAG Basics: Representing CIs with Latent and Selection variables
  11. Graphical Criteria for Markov Equivalence
  12. Algorithm: Interventional Causal Discovery under Selection
  13. Experiments and Results
  14. Simulations
  15. Real-world Applications
  16. ...and 17 more sections

Key Result

Theorem 1

For interventional distributions $\{p^{(k)}(X)\}_{k\in \{0\} \cup [K]}$ generated from DAG $\mathcal{G}$ with targets $\{I^{(k)}\}_{k\in \{0\} \cup [K]}$, let $\{ \mathcal{G}^{(I^{(k)})} \}_{k\in \{0\} \cup [K]}$ be the corresponding interventional twin graphs. For any disjoint $A,B,C \subset [D]$,

Figures (12)

  • Figure 1: Examples of the existing graph representations. (a) and (b) show mutilated DAGshauser2012characterization and the augmented DAGyang2018characterizing for $\mathcal{G} = 1\rightarrow 2\rightarrow 3$ with targets $\mathcal{I} = \{\varnothing, \{2\}, \{3\}\}$. Solid nodes represent the intervention indicators. (c) is the augmented DAG for $\mathcal{G} = 1\leftarrow L\rightarrow 2$ where $L$, in a square, is latent, and $\mathcal{I} = \{\varnothing, \{1\}\}$magliacane2016ancestral. (d) shows a seemingly natural representation for selection bias, $\mathcal{G} = 1\rightarrow S\leftarrow 2$ where $S$, in double circles, is selected, and $\mathcal{I} = \{\varnothing, \{1\}\}$. But does (d) truly capture the underlying process? See \ref{['example:clinical_X1_X2_independent']}.
  • Figure 2: (a) Scatterplot of $X_1;X_2$ in general population (both '$\times$' and '$\bullet$'), with only '$\bullet$' individuals involved into study as $p^{(0)}$. (b) and (c) show $p^{(1)}$ after two distinct but both effective interventions on $X_1$, applied to '$\bullet$' from (a).
  • Figure 3: Examples of interventional twin graphs (\ref{['def:interventional_twin_network']}). (a) and (c) are two DAGs for clinical and pest control \ref{['example:clinical_X1_X2_independent', 'example:chain_X123S']}, respectively; (b) and (d), (e) are their corresponding interventional twin graphs under different targets. The white $X$ nodes and solid $\zeta$ node are observed, forming the reality world (enclosed by solid frames), where observations or interventions are conducted. The grey nodes are unobserved, of which squares ($X^*_{\operatorname{aff}}$ and $\mathcal{E}_{\operatorname{aff}}$) are latent variables and double circles ($S^*$) are selection variables. The counterfactual basal world is enclosed by dashed frames.
  • Figure 4: Examples of MAGs of interventional twin graphs. Readers may reconstruct these MAGs from DAGs in \ref{['fig:model_define_GH_examples']} using either general rules (\ref{['def:mag_step_1_adjacencies', 'def:mag_step_2_orientations']}) or interventional twin graph-specific rules (\ref{['lem:induced_in_pk', 'lem:induced_in_target', 'lem:orient_mag']}) and verify if they match. Readers may also verify if the CI implications (\ref{['thm:i_markov_property']}) match between d-separations on DAGs and m-separations (\ref{['def:mag_m_separation']}) on MAGs.
  • Figure 5: Empirical results across different numbers of variables. Here, the first figure shows the accuracy of the edgemark of the estimated PAGs. The rest two figures show the precision and F1 score of the '$\rightarrow$' edges in PAGs. The error bars illustrate the standard errors from $20$ random simulations.
  • ...and 7 more figures

Theorems & Definitions (29)

  • Example 1
  • Example 2
  • Definition 1: Interventional twin graph
  • Theorem 1: CI and invariance implications
  • Lemma 1: Additional dependencies induced by selections
  • Lemma 2: Even more dependencies induced by interventions
  • Example 3
  • Definition 2: Markov equivalence
  • Definition 3: MAG construction step 1: adjacencies; richardson2002ancestral
  • Definition 4: MAG construction step 2: orientations; richardson2002ancestral
  • ...and 19 more