[2512.11919] A fine-grained look at causal effects in causal spaces

Statistics > Methodology arXiv:2512.11919 (stat) [Submitted on 11 Dec 2025 (v1), last revised 4 Apr 2026 (this version, v3)] Title:A fine-grained look at causal effects in causal spaces Authors:Junhyung Park, Yuqing Zhou View a PDF of the paper titled A fine-grained look at causal effects in causal spaces, by Junhyung Park and 1 other authors View PDF 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 n...