# Distance Metrics for Measuring Joint Dependence with Application to Causal Inference

arXiv: https://arxiv.org/abs/1711.09179 · HTML: https://mlsanity.com/paper/1711.09179/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2017.

## Built on by (most cited first, 7 of 7)

- [Entropy Regularized Optimal Transport Independence Criterion](https://mlsanity.com/paper/2112.15265/) (2021)
- [Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing](https://mlsanity.com/paper/2210.08149/) (2022)
- [Nystr"om M-Hilbert-Schmidt Independence Criterion](https://mlsanity.com/paper/2302.09930/) (2023)
- [Machine Learning with Multitype Protected Attributes: Intersectional Fairness through Regularisation](https://mlsanity.com/paper/2509.08163/) (2025)
- [The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels](https://mlsanity.com/paper/2403.07735/) (2024)
- [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)

Many statistical applications require the quantification of joint dependence among more than two random vectors. In this work, we generalize the notion of distance covariance to quantify joint dependence among d >= 2 random vectors. We introduce the high order distance covariance to measure the so-called Lancaster interaction dependence. The joint distance covariance is then defined as a linear combination of pairwise distance covariances and their higher order counterparts which together completely characterize mutual independence. We further introduce some related concepts including the distance cumulant, distance characteristic function, and rank-based distance covariance. Empirical estimators are constructed based on certain Euclidean distances between sample elements. We study the large sample properties of the estimators and propose a bootstrap procedure to approximate their sampling distributions. The asymptotic validity of the bootstrap procedure is justified under both the null and alternative hypotheses. The new metrics are employed to perform model selection in causal inference, which is based on the joint independence testing of the residuals from the fitted structural equation models. The effectiveness of the method is illustrated via both simulated and real datasets.