Cyclic Counterfactuals under Shift-Scale Interventions
Saptarshi Saha, Dhruv Vansraj Rathore, Utpal Garain
TL;DR
This work extends counterfactual inference to cyclic structural causal models by leveraging a global $\ell^{p}$-contraction framework. It shows that such contraction guarantees simple, uniquely solvable systems and that shift–scale interventions with bounded gains preserve this solvability in the twin-SCM construction, enabling well-defined counterfactuals even in the presence of feedback loops. The authors establish measurability and equivalence of fixed-point solution maps, demonstrate closure of shift–scale interventions under composition, and derive sub-Gaussian tail bounds for counterfactual functionals under natural regularity assumptions. An illustrative linear-Gaussian example highlights closed-form interventional and counterfactual distributions, reinforcing the practical relevance for domains with cyclic causal structure, such as biology and economics. Overall, the paper provides a mathematically principled foundation for counterfactual reasoning in cyclic SCMs and outlines directions for extending the framework to richer interventions and deep generative models.
Abstract
Most counterfactual inference frameworks traditionally assume acyclic structural causal models (SCMs), i.e. directed acyclic graphs (DAGs). However, many real-world systems (e.g. biological systems) contain feedback loops or cyclic dependencies that violate acyclicity. In this work, we study counterfactual inference in cyclic SCMs under shift-scale interventions, i.e., soft, policy-style changes that rescale and/or shift a variable's mechanism.