A fine-grained look at causal effects in causal spaces

TL;DR

This paper introduces a measure-theoretic framework for analyzing causal effects at the event level, providing new definitions and measures that extend traditional variable-based causal analysis.

Contribution

It develops a fine-grained, event-level perspective on causality within measure-theoretic causal spaces, extending existing concepts and measures.

Findings

Proposes binary definitions for the presence of causal effects.

Links causal effects to (in)dependence under intervention measures.

Recovers traditional treatment effect measures as special cases.

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.