# Sensitivity Analysis to Unobserved Confounding with Copula-based Normalizing Flows

arXiv: https://arxiv.org/abs/2508.08752 · HTML: https://mlsanity.com/paper/2508.08752/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 69 references that are in the map.
- First posted on arXiv 12 Aug 2025.

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

- [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)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [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)
- [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)
- [Deep Learning With DAGs](https://mlsanity.com/paper/2401.06864/) (2024)
- [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)
- [ρ-GNF: A Copula-based Sensitivity Analysis to Unobserved Confounding Using Normalizing Flows](https://mlsanity.com/paper/2209.07111/) (2022)

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

- [Beyond Interaction Effects: Two Logics for Studying Population Inequalities](https://mlsanity.com/paper/2601.04223/) (2025)

## Abstract (arXiv)

We propose a novel method for sensitivity analysis to unobserved confounding in causal inference. The method builds on a copula-based causal graphical normalizing flow that we term ρ-GNF, where ρ ∈ [-1,+1] is the sensitivity parameter. The parameter represents the non-causal association between exposure and outcome due to unobserved confounding, which is modeled as a Gaussian copula. In other words, the ρ-GNF enables scholars to estimate the average causal effect (ACE) as a function of ρ, accounting for various confounding strengths. The output of the ρ-GNF is what we term the ρcurve, which provides the bounds for the ACE given an interval of assumed ρ values. The ρcurve also enables scholars to identify the confounding strength required to nullify the ACE. We also propose a Bayesian version of our sensitivity analysis method. Assuming a prior over the sensitivity parameter ρ enables us to derive the posterior distribution over the ACE, which enables us to derive credible intervals. Finally, leveraging on experiments from simulated and real-world data, we show the benefits of our sensitivity analysis method.