Counterfactual identifiability beyond global monotonicity: non-monotone triangular structural causal models

TL;DR

This paper introduces non-monotone triangular structural causal models that relax global monotonicity, enabling complete counterfactual identifiability and improved performance in non-monotonic scenarios.

Contribution

It proposes a new class of causal models with mechanism-wise invertibility, proving their identifiability and demonstrating their effectiveness in synthetic and real-world tasks.

Findings

Counterfactual gains increase with non-monotonicity in synthetic data.

Perfect event-level counterfactual recovery on MuJoCo Door.

More stable recovery than Transformer and flow predictors on MuJoCo Push.

Abstract

Structural causal models provide a unified semantics for interventions and counterfactuals, but most identifiability results rely on restrictive assumptions like global monotonicity, which are often violated in embodied interaction, where the same exogenous perturbation can induce opposite responses under different contact contexts. We ask what structure still suffices once global monotonicity is dropped. We introduce non-monotone triangular structural causal models (NM-TM-SCM), which retain triangular recursion but replace global monotonicity with mechanism-wise invertibility and context-independent inverse transport. We prove that these conditions are equivalent to exogenous isomorphism and imply complete counterfactual identifiability, and we give a counterexample showing that local invertibility alone is insufficient. We instantiate the theory in CausalInverter, with triangular invertible layers, orientation gates, and transport-stability regularization. On synthetic non-monotonic mechanisms, the structural bias yields systematic counterfactual gains as non-monotonicity increases. On MuJoCo Door, our model achieves perfect event-level counterfactual recovery, lowers continuous angle error relative to a Transformer baseline, and delivers substantially more stable recovery than Transformer and conditional-flow predictors. On MuJoCo Push, where non-monotonicity is weaker, the same low-data predictors remain competitive or better, consistent with a bias-variance boundary. These results identify a broader identifiable regime between globally monotone triangular models and unconstrained black-box world models.