# Flow IV: Counterfactual Inference In Nonseparable Outcome Models Using Instrumental Variables

arXiv: https://arxiv.org/abs/2508.01321 · HTML: https://mlsanity.com/paper/2508.01321/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 2 Aug 2025.

## Builds on (most cited first, 14 of 14)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Towards Causal Representation Learning](https://mlsanity.com/paper/2102.11107/) (2021)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Theoretical Impediments to Machine Learning With Seven Sparks from the Causal Revolution](https://mlsanity.com/paper/1801.04016/) (2018)
- [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)
- [Counterfactual Image Editing](https://mlsanity.com/paper/2403.09683/) (2024)
- [Personalized Public Policy Analysis in Social Sciences using Causal-Graphical Normalizing Flows](https://mlsanity.com/paper/2202.03281/) (2022)
- [Control Function Instrumental Variable Estimation of Nonlinear Causal Effect Models](https://mlsanity.com/paper/1602.01051/) (2016)
- [Deep Learning With DAGs](https://mlsanity.com/paper/2401.06864/) (2024)
- [The Multiplicative Instrumental Variable Model](https://mlsanity.com/paper/2507.09302/) (2025)

## Built on by (most cited first, 1 of 1)

- [Overcoming Selection Bias in Statistical Studies With Amortized Bayesian Inference](https://mlsanity.com/paper/2604.18319/) (2026)

## Abstract (arXiv)

To reach human level intelligence, learning algorithms need to incorporate causal reasoning. But identifying causality, and particularly counterfactual reasoning, remains elusive. In this paper, we make progress on counterfactual inference in nonseparable outcome models by utilizing instrumental variables (IVs). IVs are a classic tool for mitigating bias from unobserved confounders when estimating causal effects. While IV methods for effect estimation have been extended to nonseparable outcome models under different assumptions, existing IV approaches to counterfactual prediction typically assume one-dimensional outcomes and additive noise. In this paper, we show that under standard IV assumptions, along with the assumption that the outcome function is invertible and has a triangular structure, then the treatment-outcome relationship becomes identifiable from observed data. We furthermore propose a method to learn the outcome function utilizing normalizing flows. This outcome function estimator can then be used to perform counterfactual inference. We refer to the method as Flow IV.