Two Layers of Instability in Causal Estimation

2026-06-19

AI Summary

Causal effect estimation from observational data often exhibits high sensitivity to minor perturbations in the data distribution, even when identifiability conditions are satisfied. This work reveals that such instability arises from two intertwined mechanisms: an intrinsic discontinuous dependence of causal effects on the data distribution, and a discontinuous aggregation of the posterior over multimodal structural causal models by common estimators. To formalize this phenomenon, we introduce the notion of “second-order instability” and develop a classification framework for estimators based on their implicit loss functions and alignment with causal effects. Drawing on decision theory, we elucidate the theoretical roots of stability differences among estimators. Our analysis demonstrates that inverse propensity weighting and regression-based estimators are discontinuous, whereas posterior mean and median estimators are continuous, thereby providing a principled foundation for designing stable causal inference methods.

Abstract

There is a precise sense in which drawing causal inferences from observational data is hard, even when identifiability is assumed. In particular, Robins and Ritov (1997) and Robins et al. (2003) showed that causal effects can be discontinuous as a function of the data distribution: two arbitrarily close data distributions might correspond to different causal effects. This is a fact independent of the choice of estimator; however, not all estimators are equally unstable. Our contribution is to surface a second layer of instability that depends on the choice of estimator. We show that many standard point estimates can be read as point summaries of multimodal distributions over the space of structural causal models. As such, estimators can jump discontinuously in the data distribution. This defines a taxonomy of estimators that admits a decision-theoretic reading: stability depends on whether the implicit loss function an estimator optimizes is aligned with the causal effect itself. Specifically, inverse propensity weighted estimators and regression estimators are examples of discontinuous summaries, while explicit posterior means and medians are shown to be continuous.

Problem

• causal estimation
• instability
• observational data
• discontinuity
• structural causal models

Innovation

• causal estimation
• instability
• structural causal models
• decision-theoretic
• posterior summarization