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A Measure-Theoretic Axiomatisation of Causality

Junhyung Park, Simon Buchholz, Bernhard Schölkopf, Krikamol Muandet

TL;DR

The paper addresses the lack of a universal axiomatisation of causality by proposing measure-theoretic causal spaces, which augment a probability space with a family of causal kernels to encode interventional information. It demonstrates that this framework can subsume interventional aspects of existing formalisms such as SCMs and PO, while naturally handling cycles, latent confounders, and continuous-time processes through explicit intervention rules like $P^{do}(U,\mathbb{Q})$ and $K^{do}(U,\mathbb{Q},L)$. Through concrete examples (confounders, cycles, Brownian motion), the authors illustrate the expressiveness and robustness of causal spaces beyond traditional models. The work provides a rigorous, decoupled foundation for causal reasoning and outlines future directions for counterfactuals and actual causality, positioning measure-theoretic causality as a complementary, formal backbone for diverse applications.

Abstract

Causality is a central concept in a wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of \textit{what happens when one intervenes on a system}, and argue in favour of taking Kolmogorov's measure-theoretic axiomatisation of probability as the starting point towards an axiomatisation of causality. To that end, we propose the notion of a \textit{causal space}, consisting of a probability space along with a collection of transition probability kernels, called \textit{causal kernels}, that encode the causal information of the space. Our proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.

A Measure-Theoretic Axiomatisation of Causality

TL;DR

The paper addresses the lack of a universal axiomatisation of causality by proposing measure-theoretic causal spaces, which augment a probability space with a family of causal kernels to encode interventional information. It demonstrates that this framework can subsume interventional aspects of existing formalisms such as SCMs and PO, while naturally handling cycles, latent confounders, and continuous-time processes through explicit intervention rules like and . Through concrete examples (confounders, cycles, Brownian motion), the authors illustrate the expressiveness and robustness of causal spaces beyond traditional models. The work provides a rigorous, decoupled foundation for causal reasoning and outlines future directions for counterfactuals and actual causality, positioning measure-theoretic causality as a complementary, formal backbone for diverse applications.

Abstract

Causality is a central concept in a wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of \textit{what happens when one intervenes on a system}, and argue in favour of taking Kolmogorov's measure-theoretic axiomatisation of probability as the starting point towards an axiomatisation of causality. To that end, we propose the notion of a \textit{causal space}, consisting of a probability space along with a collection of transition probability kernels, called \textit{causal kernels}, that encode the causal information of the space. Our proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.
Paper Structure (19 sections, 9 theorems, 25 equations, 5 figures)

This paper contains 19 sections, 9 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Causal Spaces and Interventions
  4. Comparison with Existing Frameworks
  5. Structural Causal Models (SCMs)
  6. Potential Outcomes (PO) Framework
  7. Examples
  8. Confounders
  9. Cycles
  10. Continuous-time Stochastic Processes, and Parents
  11. Conclusion
  12. Mathematical Preliminaries
  13. Measure Theory
  14. Probability Theory
  15. Causal Effect
  16. ...and 4 more sections

Key Result

Theorem 2.6

From Definition Dintervention, $\mathbb{P}^{\textnormal{do}(U,\mathbb{Q})}$ is indeed a measure on $(\Omega,\mathscr{H})$, and $\mathbb{K}^{\textnormal{do}(U,\mathbb{Q},\mathbb{L})}$ is indeed a valid causal mechanism on $(\Omega,\mathscr{H},\mathbb{P}^{\textnormal{do}(U,\mathbb{Q})})$, i.e. they sa

Figures (5)

  • Figure 1: Data generating processes and data.
  • Figure 2: Altitude and Temperature.
  • Figure 3: Correlation but no causation between ice-cream sales and shark attacks. S stands for the number of shark attacks, I for ice cream sales, T for temperature and E for economy.
  • Figure 4: Rice in the market in million tonnes and price per kg in KRW.
  • Figure 5: 1-dimensional Brownian motion, intervened and conditioned to have value 0 at time 1.

Theorems & Definitions (40)

  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 3.1: peters2017elements
  • Remark 3.2
  • Example 4.1
  • Example 4.2
  • ...and 30 more