# A simple measure of conditional dependence

arXiv: https://arxiv.org/abs/1910.12327 · HTML: https://mlsanity.com/paper/1910.12327/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2019.

## Built on by (most cited first, 30 of 38)

- [Deep Unlearning via Randomized Conditionally Independent Hessians](https://mlsanity.com/paper/2204.07655/) (2022)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [An Asymptotic Test for Conditional Independence using Analytic Kernel Embeddings](https://mlsanity.com/paper/2110.14868/) (2021)
- [Doubly Robust Conditional Independence Testing with Generative Neural Networks](https://mlsanity.com/paper/2407.17694/) (2024)
- [Ultra-marginal Feature Importance: Learning from Data with Causal Guarantees](https://mlsanity.com/paper/2204.09938/) (2022)
- [A Fast Non-parametric Approach for Local Causal Structure Learning](https://mlsanity.com/paper/2111.14969/) (2021)
- [iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models](https://mlsanity.com/paper/2306.17361/) (2023)
- [On the Self-Penalization Phenomenon in Feature Selection](https://mlsanity.com/paper/2110.05852/) (2021)
- [Learning general conditional independence structures via the neighbourhood lattice](https://mlsanity.com/paper/2206.05829/) (2022)
- [FairICP: Encouraging Equalized Odds via Inverse Conditional Permutation](https://mlsanity.com/paper/2404.05678/) (2024)
- [Limit theorems of Chatterjee's rank correlation](https://mlsanity.com/paper/2204.08031/) (2022)
- [A Novel Regularization Approach to Fair ML](https://mlsanity.com/paper/2208.06557/) (2022)
- [How Does Data Freshness Affect Real-time Supervised Learning?](https://mlsanity.com/paper/2208.06948/) (2022)
- [A survey of some recent developments in measures of association](https://mlsanity.com/paper/2211.04702/) (2022)
- [Model free variable importance for high dimensional data](https://mlsanity.com/paper/2211.08414/) (2022)
- [Assessing the overall and partial causal well-specification of nonlinear additive noise models](https://mlsanity.com/paper/2310.16502/) (2023)
- [Deep Nonparametric Conditional Independence Tests for Images](https://mlsanity.com/paper/2411.06140/) (2024)
- [Correcting Confounding via Random Selection of Background Variables](https://mlsanity.com/paper/2202.02150/) (2022)
- [Reframed GES with a Neural Conditional Dependence Measure](https://mlsanity.com/paper/2206.08531/) (2022)
- [Factor Importance Ranking and Selection using Total Indices](https://mlsanity.com/paper/2401.00800/) (2024)
- [Learning Directed Acyclic Graphs from Partial Orderings](https://mlsanity.com/paper/2403.16031/) (2024)
- [Greedy equivalence search for nonparametric graphical models](https://mlsanity.com/paper/2406.17228/) (2024)
- [Score-based Generative Modeling for Conditional Independence Testing](https://mlsanity.com/paper/2505.23309/) (2025)
- [Bivariate vine copula based regression, bivariate level and quantile curves](https://mlsanity.com/paper/2205.02557/) (2022)
- [FACT: High-Dimensional Random Forests Inference](https://mlsanity.com/paper/2207.01678/) (2022)
- [Top-k Feature Importance Ranking](https://mlsanity.com/paper/2509.15420/) (2025)
- [Causal Domain Adaptation with Copula Entropy based Conditional Independence Test](https://mlsanity.com/paper/2202.13482/) (2022)
- [Evaluating Independence and Conditional Independence Measures](https://mlsanity.com/paper/2205.07253/) (2022)
- [Estimating large causal polytrees from small samples](https://mlsanity.com/paper/2209.07028/) (2022)
- [On computing and the complexity of computing higher-order U-statistics, exactly](https://mlsanity.com/paper/2508.12627/) (2025)

## Abstract (arXiv)

We propose a coefficient of conditional dependence between two random variables Y and Z given a set of other variables X1,ldots,Xp, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in [0,1], where the limit is 0 if and only if Y and Z are conditionally independent given X1,ldots,Xp, and is 1 if and only if Y is equal to a measurable function of Z given X1,ldots,Xp. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial R2 statistic for measuring conditional dependence by regression. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.