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What is causal about causal models and representations?

Frederik Hytting Jørgensen, Luigi Gresele, Sebastian Weichwald

TL;DR

This work critically analyzes how causal models relate to real-world actions. It introduces a formal framework linking data-generating processes to causal representations via emulation and interventional validity, and shows that the natural interpretation of actions as interventions is circular, yielding universal interventional validity for correctly observational models. An impossibility result demonstrates that any non-circular interpretation satisfying natural desiderata collapses to the circular one, unless one relaxes some criteria. The paper then explores five non-circular interpretations, illustrating how they can enable falsification of causal models, and discusses broader implications for causal representation learning, causal discovery, and causal abstraction, including connections to the logic of conditionals. Overall, it argues for explicit, testable mappings between actions and interventions to ground causal representations and enable robust falsification and learning in practice.

Abstract

Causal Bayesian networks are 'causal' models since they make predictions about interventional distributions. To connect such causal model predictions to real-world outcomes, we must determine which actions in the world correspond to which interventions in the model. For example, to interpret an action as an intervention on a treatment variable, the action will presumably have to a) change the distribution of treatment in a way that corresponds to the intervention, and b) not change other aspects, such as how the outcome depends on the treatment; while the marginal distributions of some variables may change as an effect. We introduce a formal framework to make such requirements for different interpretations of actions as interventions precise. We prove that the seemingly natural interpretation of actions as interventions is circular: Under this interpretation, every causal Bayesian network that correctly models the observational distribution is trivially also interventionally valid, and no action yields empirical data that could possibly falsify such a model. We prove an impossibility result: No interpretation exists that is non-circular and simultaneously satisfies a set of natural desiderata. Instead, we examine non-circular interpretations that may violate some desiderata and show how this may in turn enable the falsification of causal models. By rigorously examining how a causal Bayesian network could be a 'causal' model of the world instead of merely a mathematical object, our formal framework contributes to the conceptual foundations of causal representation learning, causal discovery, and causal abstraction, while also highlighting some limitations of existing approaches.

What is causal about causal models and representations?

TL;DR

This work critically analyzes how causal models relate to real-world actions. It introduces a formal framework linking data-generating processes to causal representations via emulation and interventional validity, and shows that the natural interpretation of actions as interventions is circular, yielding universal interventional validity for correctly observational models. An impossibility result demonstrates that any non-circular interpretation satisfying natural desiderata collapses to the circular one, unless one relaxes some criteria. The paper then explores five non-circular interpretations, illustrating how they can enable falsification of causal models, and discusses broader implications for causal representation learning, causal discovery, and causal abstraction, including connections to the logic of conditionals. Overall, it argues for explicit, testable mappings between actions and interventions to ground causal representations and enable robust falsification and learning in practice.

Abstract

Causal Bayesian networks are 'causal' models since they make predictions about interventional distributions. To connect such causal model predictions to real-world outcomes, we must determine which actions in the world correspond to which interventions in the model. For example, to interpret an action as an intervention on a treatment variable, the action will presumably have to a) change the distribution of treatment in a way that corresponds to the intervention, and b) not change other aspects, such as how the outcome depends on the treatment; while the marginal distributions of some variables may change as an effect. We introduce a formal framework to make such requirements for different interpretations of actions as interventions precise. We prove that the seemingly natural interpretation of actions as interventions is circular: Under this interpretation, every causal Bayesian network that correctly models the observational distribution is trivially also interventionally valid, and no action yields empirical data that could possibly falsify such a model. We prove an impossibility result: No interpretation exists that is non-circular and simultaneously satisfies a set of natural desiderata. Instead, we examine non-circular interpretations that may violate some desiderata and show how this may in turn enable the falsification of causal models. By rigorously examining how a causal Bayesian network could be a 'causal' model of the world instead of merely a mathematical object, our formal framework contributes to the conceptual foundations of causal representation learning, causal discovery, and causal abstraction, while also highlighting some limitations of existing approaches.

Paper Structure

This paper contains 62 sections, 8 theorems, 47 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dialogue – What is an intervention?
  3. What went wrong?
  4. Contribution
  5. Article outline
  6. Framework
  7. Emulation and interventional validity
  8. $\mathbf{Int}_C:$ The seemingly natural interpretation of actions as interventions is circular
  9. Impossibility result for non-circular interpretations
  10. Desideratum D0:
  11. Desideratum D1:
  12. Desideratum D2:
  13. Desideratum D3:
  14. Desideratum D4:
  15. Non-circular interpretations
  16. ...and 47 more sections

Key Result

Proposition 3.2

Let a data-generating process $\mathcal{D}$, representation $\text{\boldmath$Z$}^*$, compatible CBN $\mathfrak{C}$, and set of interventions $\mathcal{I}$ in $\mathfrak{C}$ be given. Then $d\in \mathcal{I}$ is in $\mathbf{Int}^{\mathcal{I}}_C(a)$ if and only if $\mathcal{L}^{a}(\text{\boldmath$Z$}^*

Figures (3)

  • Figure 1: The framework presented in this article has three main components. 1) The observed low-level features $\text{\boldmath$X$}^*$, 2) a representation $\text{\boldmath$Z$}^*:=h(\text{\boldmath$X$}^*)$, and 3) a hypothesized causal model $\mathfrak{C}$ with variables $\text{\boldmath$Z$}$.
  • Figure 2: In \ref{['ex: TC']}, we assume that the low-level representation $(\text{LDL}^*,\text{HDL}^*,\text{HD}^*)$ is emulated by a CBN $\mathfrak{A}$, with graph given in \ref{['fig:subfigure1']}, and single-node interventions in $\mathfrak{A}$. Under $\mathbf{Int}_P$, we falsify the $(\text{TC}^*,\text{HD}^*):=(\text{LDL}^*+\text{HDL}^*,\text{HD}^*)$-compatible CBN $\mathfrak{C}$ with graph given in \ref{['fig:subfigure2']}.
  • Figure 3: Depiction of the situation in \ref{['ex: RL']}. There are $6$ different locations, three on each the left and right side $S^*$, with associated rewards $R^*$ ($+1$ or $-1$). Observationally, the stick figure picks a position $P^*$ using a uniform distribution over the $6$ locations. Whether the causal model $S\to R$ makes correct predictions about interventions on left/right, depends on which specific actions are interpreted as interventions.

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 31 more