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Kernel‐based tests for joint independence
Author(s) -
Pfister Niklas,
Bühlmann Peter,
Schölkopf Bernhard,
Peters Jonas
Publication year - 2018
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12235
Subject(s) - reproducing kernel hilbert space , mathematics , independence (probability theory) , joint probability distribution , kernel (algebra) , random variable , permutation (music) , hilbert space , statistics , discrete mathematics , pure mathematics , physics , acoustics
Summary 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.