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Meaningful Causal Aggregation and Paradoxical Confounding

Yuchen Zhu, Kailash Budhathoki, Jonas Kuebler, Dominik Janzing

TL;DR

The paper investigates how causal relations defined on aggregated macro-variables can become ill-defined or paradoxical due to micro-level realizations. It introduces the notions of natural micro-realizations and natural macro-interventions to preserve unconfoundedness under aggregation, and proposes macro-backdoor adjustment as a tool for macro-level causal inference when backdoor paths exist. Through linear Gaussian and discrete examples, it shows that aggregation can both induce and resolve confounding depending on micro-implementations, and it discusses extensions to multivariate settings and nonlinear cases. The work provides a framework to reason about macro-causal effects without requiring exact micro-model knowledge, with practical implications for policy analysis and large-scale causal inference.

Abstract

In aggregated variables the impact of interventions is typically ill-defined because different micro-realizations of the same macro-intervention can result in different changes of downstream macro-variables. We show that this ill-definedness of causality on aggregated variables can turn unconfounded causal relations into confounded ones and vice versa, depending on the respective micro-realization. We argue that it is practically infeasible to only use aggregated causal systems when we are free from this ill-definedness. Instead, we need to accept that macro causal relations are typically defined only with reference to the micro states. On the positive side, we show that cause-effect relations can be aggregated when the macro interventions are such that the distribution of micro states is the same as in the observational distribution; we term this natural macro interventions. We also discuss generalizations of this observation.

Meaningful Causal Aggregation and Paradoxical Confounding

TL;DR

The paper investigates how causal relations defined on aggregated macro-variables can become ill-defined or paradoxical due to micro-level realizations. It introduces the notions of natural micro-realizations and natural macro-interventions to preserve unconfoundedness under aggregation, and proposes macro-backdoor adjustment as a tool for macro-level causal inference when backdoor paths exist. Through linear Gaussian and discrete examples, it shows that aggregation can both induce and resolve confounding depending on micro-implementations, and it discusses extensions to multivariate settings and nonlinear cases. The work provides a framework to reason about macro-causal effects without requiring exact micro-model knowledge, with practical implications for policy analysis and large-scale causal inference.

Abstract

In aggregated variables the impact of interventions is typically ill-defined because different micro-realizations of the same macro-intervention can result in different changes of downstream macro-variables. We show that this ill-definedness of causality on aggregated variables can turn unconfounded causal relations into confounded ones and vice versa, depending on the respective micro-realization. We argue that it is practically infeasible to only use aggregated causal systems when we are free from this ill-definedness. Instead, we need to accept that macro causal relations are typically defined only with reference to the micro states. On the positive side, we show that cause-effect relations can be aggregated when the macro interventions are such that the distribution of micro states is the same as in the observational distribution; we term this natural macro interventions. We also discuss generalizations of this observation.
Paper Structure (38 sections, 9 theorems, 47 equations, 5 figures)

This paper contains 38 sections, 9 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Challenge 1. Aggregated causal relations can be ambiguous.
  3. Challenge 2. Bridging macro intervention to micro implementation.
  4. Challenge 3. The micro-model may not be known.
  5. Preliminaries: Macro- and micro-intervention
  6. Confounding on the macro level
  7. Confounding is ambiguous
  8. Macro-confounding in unconfounded micro-models
  9. A simple positive result
  10. Non-Gaussian and non-linear generalization
  11. Macro-interventions for which we cannot specify a macro confounder.
  12. Macro-confounding in confounded micro-models
  13. Linear gaussian confounded models
  14. Interpretation.
  15. Discrete confounded micro models
  16. ...and 23 more sections

Key Result

Theorem 3.1

Let $X\to Y$ and $\bar{X}$ and $\bar{Y}$ be macro-variables of $X$ and $Y$, respectively. Then the micro-realization $P^{do}_{\bar{x}}(X) \vcentcolon= P(X|\bar{X} \vcentcolon= \bar{x})$ is confounding-inhibiting, i.e. the graph $\bar{X} \to \bar{Y}$ can be drawn.

Figures (5)

  • Figure 1: Left: The micro-variable causal model $M$ is described by an SCM, which is shown within the blue frame. The aggregation map $\pi$ is applied to the micro-variables $X_1,\cdots, X_N$ and $Y_1, \cdots, Y_N$, both contained in the dashed frame. $\bar{X}$ and $\bar{Y}$ are the macro variables which arise from the aggregation map. Right: A macro-intervention $P^{do}_{\pi, \bar{x}}(X)$ is shown. The intervention applied to the macro variable $\bar{X}:= \bar{x}$ is realised as a (perfect) stochastic intervention on $X$, note that all values of $\mathbf{x}$ supported by $P_{\mathbf{X}|\bar{X}:=\bar{x}}$ give rise to $\bar{x}$.
  • Figure 2: With an appropriate microscopic micro-realizations of interventions on $\bar{X}$, the cause-effect relation (top) remains unconfounded on the macro level (lower).
  • Figure 3: Confounded cause-effect relation between $X$ and $Y$. Left: $G^a$. The amalgamated graph before macro intervention on $\bar{X}$. Right: $G^{a'}$. The amalgamated graph after macro intervention on $\bar{X}$.
  • Figure 4: The natural micro-realization of macro interventions from Section \ref{['subsec:positive']} enables coarse graining the chain (top) to the chain (lower) whenever $\bar{X}\perp \!\!\! \perp \bar{Z}\,|\bar{Y}$.
  • Figure 5: Left: complete DAG with micro variables $X,Y,Z$, together with its aggregations $\bar{X},\bar{Y},\bar{Z}$. Right: The aggregated DAG which we would like to give a causal semantics by introducing appropriate interventions, if possible.

Theorems & Definitions (27)

  • definition 1
  • definition 2: aggregation map, macro-variables
  • definition 3: amalgamated graph
  • definition 4: macro intervention
  • Remark
  • definition 5: macro-intervention graph
  • definition 6: macro confounding
  • definition 7
  • Remark
  • Theorem 3.1
  • ...and 17 more