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Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confounding

Simon Ferreira, Charles K. Assaad

TL;DR

The paper addresses causal inference in partially specified dynamic systems by focusing on macro queries in summary causal graphs (SCGs), which summarize temporal dynamics over clusters of variables and may include cycles. It shows that macro conditional independencies are identified by d-separation in SCGs, and macro total effects are identifiable via the do-calculus in SCGs, with graphical criteria (SC-projections and SC-Hedges) characterizing non-identifiability. The key contributions are the formal extension of d-separation and the do-calculus to SCGs, plus a graphical non-identifiability criterion for macro total effects. This advances causal reasoning in epidemiology and other domains where full temporal graphs are unavailable, enabling macro-level causal conclusions from partially specified spacio-temporal data under certain assumptions.

Abstract

Understanding causal relations in dynamic systems is essential in epidemiology. While causal inference methods have been extensively studied, they often rely on fully specified causal graphs, which may not always be available in complex dynamic systems. Partially specified causal graphs, and in particular summary causal graphs (SCGs), provide a simplified representation of causal relations between time series when working spacio-temporal data, omitting temporal information and focusing on causal structures between clusters of of temporal variables. Unlike fully specified causal graphs, SCGs can contain cycles, which complicate their analysis and interpretation. In addition, their cluster-based nature introduces new challenges concerning the types of queries of interest: macro queries, which involve relationships between clusters represented as vertices in the graph, and micro queries, which pertain to relationships between variables that are not directly visible through the vertices of the graph. In this paper, we first clearly distinguish between macro conditional independencies and micro conditional independencies and between macro total effects and micro total effects. Then, we demonstrate the soundness and completeness of the d-separation to identify macro conditional independencies in SCGs. Furthermore, we establish that the do-calculus is sound and complete for identifying macro total effects in SCGs. Finally, we give a graphical characterization for the non-identifiability of macro total effects in SCGs.

Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confounding

TL;DR

The paper addresses causal inference in partially specified dynamic systems by focusing on macro queries in summary causal graphs (SCGs), which summarize temporal dynamics over clusters of variables and may include cycles. It shows that macro conditional independencies are identified by d-separation in SCGs, and macro total effects are identifiable via the do-calculus in SCGs, with graphical criteria (SC-projections and SC-Hedges) characterizing non-identifiability. The key contributions are the formal extension of d-separation and the do-calculus to SCGs, plus a graphical non-identifiability criterion for macro total effects. This advances causal reasoning in epidemiology and other domains where full temporal graphs are unavailable, enabling macro-level causal conclusions from partially specified spacio-temporal data under certain assumptions.

Abstract

Understanding causal relations in dynamic systems is essential in epidemiology. While causal inference methods have been extensively studied, they often rely on fully specified causal graphs, which may not always be available in complex dynamic systems. Partially specified causal graphs, and in particular summary causal graphs (SCGs), provide a simplified representation of causal relations between time series when working spacio-temporal data, omitting temporal information and focusing on causal structures between clusters of of temporal variables. Unlike fully specified causal graphs, SCGs can contain cycles, which complicate their analysis and interpretation. In addition, their cluster-based nature introduces new challenges concerning the types of queries of interest: macro queries, which involve relationships between clusters represented as vertices in the graph, and micro queries, which pertain to relationships between variables that are not directly visible through the vertices of the graph. In this paper, we first clearly distinguish between macro conditional independencies and micro conditional independencies and between macro total effects and micro total effects. Then, we demonstrate the soundness and completeness of the d-separation to identify macro conditional independencies in SCGs. Furthermore, we establish that the do-calculus is sound and complete for identifying macro total effects in SCGs. Finally, we give a graphical characterization for the non-identifiability of macro total effects in SCGs.
Paper Structure (9 sections, 5 theorems, 4 equations, 4 figures)

This paper contains 9 sections, 5 theorems, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Summary Causal Graphs and Macro Queries
  3. Identification of Macro Conditional Independencies
  4. Identification of Macro Total Effects
  5. Strongly Connected Hedges and Non-identifiability
  6. Related Works
  7. Conclusion
  8. Acknowledgments
  9. Appendix

Key Result

Theorem 1

Let $\mathcal{G}^s=(\mathbb{S},\mathbb{E}^s)$ be an SCG and $\mathbb{X},\mathbb{Y},\mathbb{W}\subseteq \mathbb{S}$. If $\mathbb{X}$ and $\mathbb{Y}$ are d-separated by $\mathbb{W}$ in $\mathcal{G}^s$ then, in any compatible FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$, $\mathbb{V}^{\mathbb{X}}$ and

Figures (4)

  • Figure 1: An SCG with two compatible FT-ADMGs. Each pair of red and blue vertices represents the macro query we are interested in.
  • Figure 2: SCGs with identifiable macro total effects. Each pair of red and blue vertices represents the total effect we are interested in.
  • Figure 3: SCGs with not identifiable macro total effects. Each pair of red and blue vertices represents the total effect we are interested in.
  • Figure 4: SC-projections of the SCGs in Figures \ref{['fig:SCG_FTADMG']} and \ref{['fig:non_identifiable']}. Each pair of red and blue vertices represents the total effect we are interested in, and the red edges indicate those added through the SC-projection.

Theorems & Definitions (27)

  • Definition 1: Discrete-time dynamic structural causal model (DTDSCM)
  • Definition 2: Full-Time Acyclic Directed Mixed Graph
  • Definition 3: Summary Causal Graph with possible latent confounding
  • Definition 4: Micro conditional independency
  • Definition 5: Macro conditional independency
  • Definition 6: Micro total effect
  • Definition 7: Macro total effect
  • Definition 8: d-separation in SCGs
  • proof
  • Theorem 1: Soundness of d-separation in SCGs
  • ...and 17 more