# Kernel-based Tests for Joint Independence

arXiv: https://arxiv.org/abs/1603.00285 · HTML: https://mlsanity.com/paper/1603.00285/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Mar 2016.

## Built on by (most cited first, 30 of 40)

- [Towards Causal Representation Learning](https://mlsanity.com/paper/2102.11107/) (2021)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [From Statistical to Causal Learning](https://mlsanity.com/paper/2204.00607/) (2022)
- [Exploiting Independent Instruments: Identification and Distribution Generalization](https://mlsanity.com/paper/2202.01864/) (2022)
- [Grouped Feature Importance and Combined Features Effect Plot](https://mlsanity.com/paper/2104.11688/) (2021)
- [Backward-Compatible Prediction Updates: A Probabilistic Approach](https://mlsanity.com/paper/2107.01057/) (2021)
- [Variance Covariance Regularization Enforces Pairwise Independence in Self-Supervised Representations](https://mlsanity.com/paper/2209.14905/) (2022)
- [Entropy Regularized Optimal Transport Independence Criterion](https://mlsanity.com/paper/2112.15265/) (2021)
- [Cause-Effect Inference in Location-Scale Noise Models: Maximum Likelihood vs. Independence Testing](https://mlsanity.com/paper/2301.12930/) (2023)
- [Neural Mixed Effects for Nonlinear Personalized Predictions](https://mlsanity.com/paper/2306.08149/) (2023)
- [Kernelized Cumulants: Beyond Kernel Mean Embeddings](https://mlsanity.com/paper/2301.12466/) (2023)
- [Nystr"om M-Hilbert-Schmidt Independence Criterion](https://mlsanity.com/paper/2302.09930/) (2023)
- [Structural restrictions in local causal discovery: identifying direct causes of a target variable](https://mlsanity.com/paper/2307.16048/) (2023)
- [Parameter identification in linear non-Gaussian causal models under general confounding](https://mlsanity.com/paper/2405.20856/) (2024)
- [A Simple Unified Approach to Testing High-Dimensional Conditional Independences for Categorical and Ordinal Data](https://mlsanity.com/paper/2206.04356/) (2022)
- [Multi-Domain Causal Discovery in Bijective Causal Models](https://mlsanity.com/paper/2504.21261/) (2025)
- [BEAUTY Powered BEAST](https://mlsanity.com/paper/2103.00674/) (2021)
- [A survey of some recent developments in measures of association](https://mlsanity.com/paper/2211.04702/) (2022)
- [Causal Discovery via Conditional Independence Testing with Proxy Variables](https://mlsanity.com/paper/2305.05281/) (2023)
- [Kernel-based Joint Independence Tests for Multivariate Stationary and Non-stationary Time Series](https://mlsanity.com/paper/2305.08529/) (2023)
- [Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions](https://mlsanity.com/paper/2306.00904/) (2023)
- [Model-based causal feature selection for general response types](https://mlsanity.com/paper/2309.12833/) (2023)
- [Assessing the overall and partial causal well-specification of nonlinear additive noise models](https://mlsanity.com/paper/2310.16502/) (2023)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Data-Driven Representations for Testing Independence: Modeling, Analysis and Connection with Mutual Information Estimation](https://mlsanity.com/paper/2110.14122/) (2021)
- [The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels](https://mlsanity.com/paper/2403.07735/) (2024)
- [Kernel Density Machines](https://mlsanity.com/paper/2504.21419/) (2025)
- [Detecting Unobserved Confounders: A Kernelized Regression Approach](https://mlsanity.com/paper/2601.00200/) (2026)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Permutation-Free High-Order Interaction Tests](https://mlsanity.com/paper/2506.05963/) (2025)

## Abstract (arXiv)

We investigate the problem of testing whether d random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the d-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the d-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the d variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small d, it performs well in practice. Finally, we apply the test to a problem in causal discovery.