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

arXiv: https://arxiv.org/abs/2605.04413 · HTML: https://mlsanity.com/paper/2605.04413/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 33 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 21 of 21)

- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [Mastering Atari with Discrete World Models](https://mlsanity.com/paper/2010.02193/) (2020)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Deep Structural Causal Models for Tractable Counterfactual Inference](https://mlsanity.com/paper/2006.06485/) (2020)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Neural Causal Models for Counterfactual Identification and Estimation](https://mlsanity.com/paper/2210.00035/) (2022)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Graphical Normalizing Flows](https://mlsanity.com/paper/2006.02548/) (2020)
- [PIVOT-R: Primitive-Driven Waypoint-Aware World Model for Robotic Manipulation](https://mlsanity.com/paper/2410.10394/) (2024)
- [Counterfactual Generation with Identifiability Guarantees](https://mlsanity.com/paper/2402.15309/) (2024)
- [Smoke and Mirrors in Causal Downstream Tasks](https://mlsanity.com/paper/2405.17151/) (2024)
- [Marginal Causal Flows for Validation and Inference](https://mlsanity.com/paper/2411.01295/) (2024)
- [On the Parameter Identifiability of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2407.16975/) (2024)
- [Consistency of Neural Causal Partial Identification](https://mlsanity.com/paper/2405.15673/) (2024)
- [Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models](https://mlsanity.com/paper/2306.11281/) (2023)
- [Conditional Generative Models are Sufficient to Sample from Any Causal Effect Estimand](https://mlsanity.com/paper/2402.07419/) (2024)
- [Intervention and Conditioning in Causal Bayesian Networks](https://mlsanity.com/paper/2405.14728/) (2024)
- [Exogenous Isomorphism for Counterfactual Identifiability](https://mlsanity.com/paper/2505.02212/) (2025)
- [Exogenous Matching: Learning Good Proposals for Tractable Counterfactual Estimation](https://mlsanity.com/paper/2410.13914/) (2024)

## Abstract (arXiv)

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.