# A fine-grained look at causal effects in causal spaces

arXiv: https://arxiv.org/abs/2512.11919 · HTML: https://mlsanity.com/paper/2512.11919/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 13 of 13)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Causal Abstraction: A Theoretical Foundation for Mechanistic Interpretability](https://mlsanity.com/paper/2301.04709/) (2023)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Locally Typical Sampling](https://mlsanity.com/paper/2202.00666/) (2022)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Conditional Distributional Treatment Effect with Kernel Conditional Mean Embeddings and U-Statistic Regression](https://mlsanity.com/paper/2102.08208/) (2021)
- [A Measure-Theoretic Characterization of Tight Language Models](https://mlsanity.com/paper/2212.10502/) (2022)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [Robust and Agnostic Learning of Conditional Distributional Treatment Effects](https://mlsanity.com/paper/2205.11486/) (2022)
- [A Measure-Theoretic Axiomatisation of Causality](https://mlsanity.com/paper/2305.17139/) (2023)
- [Formal Aspects of Language Modeling](https://mlsanity.com/paper/2311.04329/) (2023)
- [Generalization Bounds via Conditional f-Information](https://mlsanity.com/paper/2410.22887/) (2024)

## Abstract (arXiv)

The notion of causal effect is fundamental across many scientific disciplines. Traditionally, quantitative researchers have studied causal effects at the level of variables; for example, how a certain drug dose (W) causally affects a patient's blood pressure (Y). However, in many modern data domains, the raw variables-such as pixels in an image or tokens in a language model-do not have the semantic structure needed to formulate meaningful causal questions. In this paper, we offer a more fine-grained perspective by studying causal effects at the level of events, drawing inspiration from probability theory, where core notions such as independence are first given for events and sigma-algebras, before random variables enter the picture. Within the measure-theoretic framework of causal spaces, a recently introduced axiomatisation of causality, we first introduce several binary definitions that determine whether a causal effect is present, as well as proving some properties of them linking causal effect to (in)dependence under an intervention measure. Further, we provide quantifying measures that capture the strength and nature of causal effects on events, and show that we can recover the common measures of treatment effect as special cases.