The Causal Uncertainty Principle

TL;DR

The paper tackles the enduring tension between internal and external validity by positing a structural mechanism: evidential states that are transformed by three basic study operations (restriction, conditioning, intervention) in a non-commutative manner. It formalises how order-dependent transformations sharpen causal identification but erode heterogeneity needed for generalisability, introducing a causal-breadth constraint to capture this trade-off. Through observational and experimental examples, it explains common transportability failures and reframes study-design evaluation as a sequence-sensitive process. The framework provides conceptual clarity for choosing design choices based on whether generalisability or precise causal identification is the priority.

Abstract

This paper explains why internal and external validity cannot be simultaneously maximised. It introduces "evidential states" to represent the information available for causal inference and shows that routine study operations (restriction, conditioning, and intervention) transform these states in ways that do not commute. Because each operation removes or reorganises information differently, changing their order yields evidential states that support different causal claims. This non-commutativity creates a structural trade-off: the steps that secure precise causal identification also eliminate the heterogeneity required for generalisation. Small model, observational and experimental examples illustrate how familiar failures of transportability arise from this order dependence. The result is a concise structural account of why increasing causal precision necessarily narrows the world to which findings apply.

Figures (2)

• Figure 1: Minimal observational causal diagram illustrating non-commutativity of study operations. An unobserved common cause ($U$) influences both treatment ($T$) and outcome ($Y$). An observed variable ($X$), correlated with $U$, provides a path for adjustment when its variation is preserved. Conditioning on $X$ before restricting the population blocks the confounding path, but restricting first can eliminate variation in $X$, making adjustment impossible. The two sequences of operations lead to different evidential states and incompatible causal conclusions.
• Figure 2: Minimal experimental causal diagram showing that intervention does not escape evidential non-commutativity. Eligibility criteria ($R$) restrict who is eligible for randomisation ($I$), and post-randomisation restrictions ($R'$)—such as attrition or protocol deviations—determine which treatment assignments ($T$) remain in the analytic sample. Randomisation ensures internal validity within this restricted evidential world, but the restrictions before and after intervention create divergence from the target population. Thus restriction and intervention do not commute, and experimental estimates obey the same causal-generalisability trade-off as observational designs.