A fine-grained look at causal effects in causal spaces

Junhyung Park, Yuqing Zhou
Proceedings of the Fifth Conference on Causal Learning and Reasoning, PMLR 323:166-195, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v323-park26a, title = {A fine-grained look at causal effects in causal spaces}, author = {Park, Junhyung and Zhou, Yuqing}, booktitle = {Proceedings of the Fifth Conference on Causal Learning and Reasoning}, pages = {166--195}, year = {2026}, editor = {Mazaheri, Bijan and Hanson, Niels Richard}, volume = {323}, series = {Proceedings of Machine Learning Research}, month = {06--08 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v323/main/assets/park26a/park26a.pdf}, url = {https://proceedings.mlr.press/v323/park26a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T A fine-grained look at causal effects in causal spaces %A Junhyung Park %A Yuqing Zhou %B Proceedings of the Fifth Conference on Causal Learning and Reasoning %C Proceedings of Machine Learning Research %D 2026 %E Bijan Mazaheri %E Niels Richard Hanson %F pmlr-v323-park26a %I PMLR %P 166--195 %U https://proceedings.mlr.press/v323/park26a.html %V 323 %X 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.
APA
Park, J. & Zhou, Y.. (2026). A fine-grained look at causal effects in causal spaces. Proceedings of the Fifth Conference on Causal Learning and Reasoning, in Proceedings of Machine Learning Research 323:166-195 Available from https://proceedings.mlr.press/v323/park26a.html.

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