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Kernel‐based tests for joint independence

Author

Listed:
  • Niklas Pfister
  • Peter Bühlmann
  • Bernhard Schölkopf
  • Jonas Peters

Abstract

We investigate the problem of testing whether d possibly multivariate 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 but allows for an arbitrary number of variables. We embed the joint distribution and the product of the marginals in 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 0 if and only if the d variables are jointly independent, as long as the kernel is characteristic. On the basis of an empirical estimate of dHSIC, we investigate three non‐parametric hypothesis tests: a permutation test, a bootstrap analogue and a procedure based on a gamma approximation. We apply non‐parametric independence testing to a problem in causal discovery and illustrate the new methods on simulated and real data sets.

Suggested Citation

  • Niklas Pfister & Peter Bühlmann & Bernhard Schölkopf & Jonas Peters, 2018. "Kernel‐based tests for joint independence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(1), pages 5-31, January.
    • Handle: RePEc:bla:jorssb:v:80:y:2018:i:1:p:5-31
    DOI: 10.1111/rssb.12235
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    Cited by:

    1. Rafael Carvalho Ceregatti & Rafael Izbicki & Luis Ernesto Bueno Salasar, 2021. "WIKS: a general Bayesian nonparametric index for quantifying differences between two populations," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 274-291, March.
    2. S Gorsky & L Ma, 2022. "Multi-scale Fisher’s independence test for multivariate dependence [A simple measure of conditional dependence]," Biometrika, Biometrika Trust, vol. 109(3), pages 569-587.
    3. Chaudhuri, Arin & Hu, Wenhao, 2019. "A fast algorithm for computing distance correlation," Computational Statistics & Data Analysis, Elsevier, vol. 135(C), pages 15-24.
    4. Roy, Angshuman & Ghosh, Anil K., 2020. "Some tests of independence based on maximum mean discrepancy and ranks of nearest neighbors," Statistics & Probability Letters, Elsevier, vol. 164(C).
    5. Zhang, Jin-Ting & Zhu, Tianming, 2024. "A fast and accurate kernel-based independence test with applications to high-dimensional and functional data," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    6. Lai, Tingyu & Zhang, Zhongzhan & Wang, Yafei & Kong, Linglong, 2021. "Testing independence of functional variables by angle covariance," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    7. Zhang, Qingyang, 2019. "Independence test for large sparse contingency tables based on distance correlation," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 17-22.
    8. Zhang, Qingyang, 2023. "On the asymptotic null distribution of the symmetrized Chatterjee’s correlation coefficient," Statistics & Probability Letters, Elsevier, vol. 194(C).
    9. Xu, Kai & Cheng, Qing, 2024. "Test of conditional independence in factor models via Hilbert–Schmidt independence criterion," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    10. Lai, Tingyu & Zhang, Zhongzhan & Wang, Yafei, 2021. "A kernel-based measure for conditional mean dependence," Computational Statistics & Data Analysis, Elsevier, vol. 160(C).
    11. Linda Chamakh & Zoltan Szabo, 2021. "Keep it Tighter -- A Story on Analytical Mean Embeddings," Papers 2110.09516, arXiv.org, revised Nov 2024.
    12. Fernández-Durán Juan José & Gregorio-Domínguez María Mercedes, 2023. "Test of bivariate independence based on angular probability integral transform with emphasis on circular-circular and circular-linear data," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-17, January.
    13. repec:ehl:lserod:122819 is not listed on IDEAS
    14. Chamakh, Linda & Szabo, Zoltan, 2021. "Kernel minimum divergence portfolios," LSE Research Online Documents on Economics 115723, London School of Economics and Political Science, LSE Library.
    15. Zhang, Wei & Gao, Wei & Ng, Hon Keung Tony, 2023. "Multivariate tests of independence based on a new class of measures of independence in Reproducing Kernel Hilbert Space," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    16. Kuang‐Yao Lee & Lexin Li, 2022. "Functional structural equation model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 600-629, April.

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