A simpler condition for consistency of a kernel independence test
By Arthur Gretton. 2015
25 citations recorded by the source corpus.
A statistical test of independence may be constructed using the Hilbert-Schmidt Independence Criterion (HSIC) as a test statistic. The HSIC is defined as the distance between the embedding of the joint distribution, and the embedding of the product of the marginals, in a Reproducing Kernel Hilbert Space (RKHS). It has previously been shown that when the kernel used in defining the joint embedding is characteristic (that is, the embedding of the joint distribution to the feature space is injective), then the HSIC-based test is consistent. In particular, it is sufficient for the product of kernels on the individual domains to be characteristic on the joint domain. In this note, it is established via a result of Lyons (2013) that HSIC-based independence tests are consistent when kernels on the marginals are characteristic on their respective domains, even when the product of kernels is not characteristic on the joint domain.
Subfields: Statistics and Probability, Statistical Learning Theory
How researchers see this paper
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In Statistics and Probability
- Bernhard Schölkopf
- Jan Peters
- Dominik Janzing
- Arthur Gretton
- Dino Sejdinovic
- Peter Buhlmann
- Joris M. Mooij
- Niklas Pfister
In Statistical Learning Theory
- Bernhard Schölkopf
- Arthur Gretton
- Dino Sejdinovic
- Jan Peters
- Kacper P. Chwialkowski
- Dominik Janzing
- Kenji Fukumizu
- Bharath K. Sriperumbudur
Citations in the corpus
- Kernel‐based tests for joint independence (2016, methodology): “Such a mapping can in fact be seen as a generalization of the L(2) distance between “traditional” kernel density estimators, see the discussion on page 732 in (Gretton et al., 2012). For characteristic kernels (e.g., the popular Gaussian kernel), the embedding of Borel probability measures is inject”
- Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “Recently, Gretton (2015) showed that a similar result also holds if both kernels k and l are characteristic and satisfy some other conditions as well.”
- A Wild Bootstrap for Degenerate Kernel Tests (2014, methodology): “When characteristic RKHSs are used, the HSIC is zero iff X |= Y : this follows from [18].”
- A low variance consistent test of relative dependency (2014, background): “HSIC determines independence: HSIC = 0 iff Pxy = PxPy when kernels k and l are characteristic on their respective marginal domains (Gretton, 2015).”