# Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model

arXiv: https://arxiv.org/abs/2306.01424 · HTML: https://mlsanity.com/paper/2306.01424/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 91 of its 153 references that are in the map.
- First posted on arXiv 2 Jun 2023.

## Builds on (most cited first, 30 of 91)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Counterfactual Multi-Agent Policy Gradients](https://mlsanity.com/paper/1705.08926/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Causal Effect Inference with Deep Latent-Variable Models](https://mlsanity.com/paper/1705.08821/) (2017)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Counterfactual Visual Explanations](https://mlsanity.com/paper/1904.07451/) (2019)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [Deep Structural Causal Models for Tractable Counterfactual Inference](https://mlsanity.com/paper/2006.06485/) (2020)
- [CausalGAN: Learning Causal Implicit Generative Models with Adversarial Training](https://mlsanity.com/paper/1709.02023/) (2017)
- [Nonparametric Estimation of Heterogeneous Treatment Effects: From Theory to Learning Algorithms](https://mlsanity.com/paper/2101.10943/) (2021)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [A survey of algorithmic recourse: definitions, formulations, solutions, and prospects](https://mlsanity.com/paper/2010.04050/) (2020)
- [Counterfactual Off-Policy Evaluation with Gumbel-Max Structural Causal Models](https://mlsanity.com/paper/1905.05824/) (2019)
- [Counterfactual Generative Networks](https://mlsanity.com/paper/2101.06046/) (2021)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [The Causal-Neural Connection: Expressiveness, Learnability, and Inference](https://mlsanity.com/paper/2107.00793/) (2021)
- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [Diffusion Causal Models for Counterfactual Estimation](https://mlsanity.com/paper/2202.10166/) (2022)
- [The Incomplete Rosetta Stone Problem: Identifiability Results for Multi-View Nonlinear ICA](https://mlsanity.com/paper/1905.06642/) (2019)
- [Woulda, Coulda, Shoulda: Counterfactually-Guided Policy Search](https://mlsanity.com/paper/1811.06272/) (2018)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)

## Built on by (most cited first, 14 of 14)

- [A Neural Framework for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2311.16026/) (2023)
- [Causal Fairness under Unobserved Confounding: A Neural Sensitivity Framework](https://mlsanity.com/paper/2311.18460/) (2023)
- [Quantifying Aleatoric Uncertainty of the Treatment Effect: A Novel Orthogonal Learner](https://mlsanity.com/paper/2411.03387/) (2024)
- [Learning Structural Causal Models through Deep Generative Models: Methods, Guarantees, and Challenges](https://mlsanity.com/paper/2405.05025/) (2024)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [Consistency of Neural Causal Partial Identification](https://mlsanity.com/paper/2405.15673/) (2024)
- [Consistent End-to-End Estimation for Counterfactual Fairness](https://mlsanity.com/paper/2310.17687/) (2023)
- [Targeted Sequential Indirect Experiment Design](https://mlsanity.com/paper/2405.19985/) (2024)
- [Your Assumed DAG is Wrong and Here's How To Deal With It](https://mlsanity.com/paper/2502.17030/) (2025)
- [Causal Inference on Networks under Misspecified Exposure Mappings: A Partial Identification Framework](https://mlsanity.com/paper/2602.03459/) (2026)
- [Learning Counterfactual Outcomes Under Rank Preservation](https://mlsanity.com/paper/2502.06398/) (2025)
- [Learning Counterfactually Fair Models via Improved Generation with Neural Causal Models](https://mlsanity.com/paper/2502.12796/) (2025)
- [Testing for Causal Fairness](https://mlsanity.com/paper/2502.12874/) (2025)
- [ConfoundingSHAP: Quantifying confounding strength in causal inference](https://mlsanity.com/paper/2605.10533/) (2026)

## Abstract (arXiv)

Counterfactual inference aims to answer retrospective "what if" questions and thus belongs to the most fine-grained type of inference in Pearl's causality ladder. Existing methods for counterfactual inference with continuous outcomes aim at point identification and thus make strong and unnatural assumptions about the underlying structural causal model. In this paper, we relax these assumptions and aim at partial counterfactual identification of continuous outcomes, i.e., when the counterfactual query resides in an ignorance interval with informative bounds. We prove that, in general, the ignorance interval of the counterfactual queries has non-informative bounds, already when functions of structural causal models are continuously differentiable. As a remedy, we propose a novel sensitivity model called Curvature Sensitivity Model. This allows us to obtain informative bounds by bounding the curvature of level sets of the functions. We further show that existing point counterfactual identification methods are special cases of our Curvature Sensitivity Model when the bound of the curvature is set to zero. We then propose an implementation of our Curvature Sensitivity Model in the form of a novel deep generative model, which we call Augmented Pseudo-Invertible Decoder. Our implementation employs (i) residual normalizing flows with (ii) variational augmentations. We empirically demonstrate the effectiveness of our Augmented Pseudo-Invertible Decoder. To the best of our knowledge, ours is the first partial identification model for Markovian structural causal models with continuous outcomes.