Kernel‐based tests for joint independence

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.