# The Weighted Generalised Covariance Measure

arXiv: https://arxiv.org/abs/2111.04361 · HTML: https://mlsanity.com/paper/2111.04361/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2021.

## Built on by (most cited first, 13 of 13)

- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [Nonparametric Conditional Local Independence Testing](https://mlsanity.com/paper/2203.13559/) (2022)
- [On the Hardness of Conditional Independence Testing In Practice](https://mlsanity.com/paper/2512.14000/) (2025)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [General Frameworks for Conditional Two-Sample Testing](https://mlsanity.com/paper/2410.16636/) (2024)
- [Deep Nonparametric Conditional Independence Tests for Images](https://mlsanity.com/paper/2411.06140/) (2024)
- [A general framework for the analysis of kernel-based tests](https://mlsanity.com/paper/2209.00124/) (2022)
- [The Generalised Kernel Covariance Measure](https://mlsanity.com/paper/2604.03721/) (2026)
- [Effect-Invariant Mechanisms for Policy Generalization](https://mlsanity.com/paper/2306.10983/) (2023)
- [Evaluating Independence and Conditional Independence Measures](https://mlsanity.com/paper/2205.07253/) (2022)
- [Differentially Private Conditional Independence Testing](https://mlsanity.com/paper/2306.06721/) (2023)
- [Conditional Independence Tests for Constraint-Based Causal Discovery: A Survey](https://mlsanity.com/paper/2608.11156/) (2026)
- [GFCM: A Tail-Sensitive Mixed-Type Conditional Independence Test for Causal Discovery](https://mlsanity.com/paper/2608.15332/) (2026)

## Abstract (arXiv)

We introduce a new test for conditional independence which is based on what we call the weighted generalised covariance measure (WGCM). It is an extension of the recently introduced generalised covariance measure (GCM). To test the null hypothesis of X and Y being conditionally independent given Z, our test statistic is a weighted form of the sample covariance between the residuals of nonlinearly regressing X and Y on Z. We propose different variants of the test for both univariate and multivariate X and Y . We give conditions under which the tests yield the correct type I error rate. Finally, we compare our novel tests to the original GCM using simulation and on real data sets. Typically, our tests have power against a wider class of alternatives compared to the GCM. This comes at the cost of having less power against alternatives for which the GCM already works well. In the special case of binary or categorical X and Y , one of our tests has power against all alternatives.