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Causal Effect Identification in a Sub-Population with Latent Variables

Amir Mohammad Abouei, Ehsan Mokhtarian, Negar Kiyavash, Matthias Grossglauser

TL;DR

This paper proposes a sound algorithm for the s-ID problem with latent variables, and extends the classical relevant graphical definitions, such as c-components and Hedges, initially defined for the so-called ID problem to their new counterparts.

Abstract

The s-ID problem seeks to compute a causal effect in a specific sub-population from the observational data pertaining to the same sub population (Abouei et al., 2023). This problem has been addressed when all the variables in the system are observable. In this paper, we consider an extension of the s-ID problem that allows for the presence of latent variables. To tackle the challenges induced by the presence of latent variables in a sub-population, we first extend the classical relevant graphical definitions, such as c-components and Hedges, initially defined for the so-called ID problem (Pearl, 1995; Tian & Pearl, 2002), to their new counterparts. Subsequently, we propose a sound algorithm for the s-ID problem with latent variables.

Causal Effect Identification in a Sub-Population with Latent Variables

TL;DR

This paper proposes a sound algorithm for the s-ID problem with latent variables, and extends the classical relevant graphical definitions, such as c-components and Hedges, initially defined for the so-called ID problem to their new counterparts.

Abstract

The s-ID problem seeks to compute a causal effect in a specific sub-population from the observational data pertaining to the same sub population (Abouei et al., 2023). This problem has been addressed when all the variables in the system are observable. In this paper, we consider an extension of the s-ID problem that allows for the presence of latent variables. To tackle the challenges induced by the presence of latent variables in a sub-population, we first extend the classical relevant graphical definitions, such as c-components and Hedges, initially defined for the so-called ID problem (Pearl, 1995; Tian & Pearl, 2002), to their new counterparts. Subsequently, we propose a sound algorithm for the s-ID problem with latent variables.
Paper Structure (13 sections, 14 theorems, 74 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 14 theorems, 74 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ID, c-component, and Hedge
  4. s-ID, s-component, and s-Hedge
  5. Main Results
  6. Reduction from s-Recoverability to s-ID
  7. Conclusion
  8. Additional Example
  9. Technical Preliminaries
  10. Proofs of Main Results
  11. Numerical Experiment
  12. Theoretical Analysis
  13. Empirical Analysis

Key Result

Theorem 3.4

Let $\mathcal{G}$ be an ADMG over $\mathbf{V}$, and $\mathbf{X}$ and $\mathbf{Y}$ be two disjoint subsets of $\mathbf{V}$. Causal effect $P_{\mathbf{X}}(\mathbf{Y})$ is ID in $\mathcal{G}$ if and only if $Q[\mathbf{D}]$ is ID in $\mathcal{G}$, where $\mathbf{D} = \textit{Anc}_{\mathcal{G}[\mathbf{V}

Figures (8)

  • Figure 1: A population consists of a sample space for the study of the causal effect of an intervention. While the unbiased sampler draws samples uniformly at random, the biased sampler selects samples based on certain criteria, forming a sub-population.
  • Figure 2: ADMG $\mathcal{G}^{\textsc{s}}$ in Example 1.
  • Figure 3: ADMGs in Examples of Sections \ref{['sec: ID']}, \ref{['sec: s-ID']}, and \ref{['sec: main']}.
  • Figure 4: Visualization of the graph structures defined in Sections \ref{['sec: ID']} and \ref{['sec: s-ID']}.
  • Figure 5: The augmented ADMG $\mathcal{G}^{\textsc{s}}$ in Example \ref{['ex: ex10']}.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Definition 3.1: ID
  • Definition 3.2: c-component
  • Definition 3.3: Hedge
  • Example 2
  • Theorem 3.4: ID
  • Example 3
  • Definition 4.1: s-ID
  • Definition 4.2: $Q^\textsc{s}{[\cdot]}$
  • Definition 4.3: s-component
  • Example 4
  • ...and 30 more