# ρ-GNF: A Copula-based Sensitivity Analysis to Unobserved Confounding Using Normalizing Flows

arXiv: https://arxiv.org/abs/2209.07111 · HTML: https://mlsanity.com/paper/2209.07111/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 80 references that are in the map.
- First posted on arXiv 15 Sep 2022.

## Builds on (most cited first, 13 of 13)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Graphical Normalizing Flows](https://mlsanity.com/paper/2006.02548/) (2020)
- [Sense and Sensitivity Analysis: Simple Post-Hoc Analysis of Bias Due to Unobserved Confounding](https://mlsanity.com/paper/2003.01747/) (2020)
- [Deep Archimedean Copulas](https://mlsanity.com/paper/2012.03137/) (2020)
- [Personalized Public Policy Analysis in Social Sciences using Causal-Graphical Normalizing Flows](https://mlsanity.com/paper/2202.03281/) (2022)
- [Copula-based Sensitivity Analysis for Multi-Treatment Causal Inference with Unobserved Confounding](https://mlsanity.com/paper/2102.09412/) (2021)
- [Simple yet Sharp Sensitivity Analysis for Unmeasured Confounding](https://mlsanity.com/paper/2104.13020/) (2021)
- [Sensitivity to Unobserved Confounding in Studies with Factor-structured Outcomes](https://mlsanity.com/paper/2208.06552/) (2022)
- [Counterfactual Analysis of the Impact of the IMF Program on Child Poverty in the Global-South Region using Causal-Graphical Normalizing Flows](https://mlsanity.com/paper/2202.09391/) (2022)

## Built on by (most cited first, 2 of 2)

- [Sensitivity Analysis to Unobserved Confounding with Copula-based Normalizing Flows](https://mlsanity.com/paper/2508.08752/) (2025)
- [Beyond Interaction Effects: Two Logics for Studying Population Inequalities](https://mlsanity.com/paper/2601.04223/) (2025)

## Abstract (arXiv)

We propose a novel sensitivity analysis to unobserved confounding in observational studies using copulas and normalizing flows. Using the idea of interventional equivalence of structural causal models, we develop ρ-GNF (ρ-graphical normalizing flow), where ρ∈[-1,+1] is a bounded sensitivity parameter. This parameter represents the back-door non-causal association due to unobserved confounding, and which is encoded with a Gaussian copula. In other words, the ρ-GNF enables scholars to estimate the average causal effect (ACE) as a function of ρ, while accounting for various assumed strengths of the unobserved confounding. The output of the ρ-GNF is what we denote as the ρcurve that provides the bounds for the ACE given an interval of assumed ρ values. In particular, the ρcurve enables scholars to identify the confounding strength required to nullify the ACE, similar to other sensitivity analysis methods (e.g., the E-value). Leveraging on experiments from simulated and real-world data, we show the benefits of ρ-GNF. One benefit is that the ρ-GNF uses a Gaussian copula to encode the distribution of the unobserved causes, which is commonly used in many applied settings. This distributional assumption produces narrower ACE bounds compared to other popular sensitivity analysis methods.