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  Discussion of: Brownian distance covariance

Gretton, A., Fukumizu, K., & Sriperumbudur, B. (2009). Discussion of: Brownian distance covariance. The Annals of Applied Statistics, 3(4), 1285-1294. doi:10.1214/09-AOAS312E.

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Gretton, A1, Author              
Fukumizu, K1, Author              
Sriperumbudur, BK1, Author              
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1Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society, ou_1497795              

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 Abstract: A dependence statistic, the Brownian Distance Covariance, has been proposed for use in dependence measurement and independence testing: we refer to this contribution henceforth as SR [we also note the earlier work on this topic of Székely, Rizzo and Bakirov (2007)]. Some advantages of the authors’ approach are that the random variables X and Y being tested may have arbitrary di- mension Rp and Rq , respectively; and the test is consistent against all alternatives subject to the conditions E‖X‖p < ∞ and E‖X‖q < ∞. In our discussion we review and compare against a number of related depen- dence measures that have appeared in the statistics and machine learning litera- ture. We begin with distances of the form of SR, equation (2.2), most notably the work of Feuerverger (1993); Kankainen (1995); Kankainen and Ushakov (1998); Ushakov (1999), which we describe in Section 2: these measures have been for- mulated only for the case p = q = 1, however. In Section 3 we turn to more recent dependence measures which are computed between mappings of the probability distributions Px , Py , and Pxy of X, Y , and (X, Y ), respectively, to high dimen- sional feature spaces: specifically, reproducing kernel Hilbert spaces (RKHSs). The RKHS dependence statistics may be based on the distance [Smola et al. (2007), Section 2.3], covariance [Gretton et al. (2005a, 2005b, 2008)], or corre- lation [Dauxois and Nkiet (1998); Bach and Jordan (2002); Fukumizu, Bach and Gretton (2007); Fukumizu et al. (2008)] between the feature mappings, and make smoothness assumptions which can improve the power of the tests over approaches relying on distances between the unmapped variables. When the RKHSs are char- acteristic [Fukumizu et al. (2008); Sriperumbudur et al. (2008)], meaning that the feature mapping from the space of probability measures to the RKHS is injective, the kernel-based tests are consistent for all probability measures generating (X, Y ). RKHS-based tests apply on spaces Rp × Rq for arbitrary p and q. In fact, kernel independence tests are applicable on a still broader range of (possibly non-Euclidean) domains, which can include strings [Leslie et al. (2002)], graphs [Gärtner, Flach and Wrobel (2003)], and groups [Fukumizu et al. (2009)], making the kernel approach very general. In Section 4 we provide an empirical comparison between the approach of SR and the kernel statistic of Gretton et al. (2005b, 2008) on an independence testing benchmark.

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 Dates: 2009-12
 Publication Status: Published in print
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 Identifiers: DOI: 10.1214/09-AOAS312E
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Title: The Annals of Applied Statistics
  Other : The Annals of Applied Statistics: An official journal of the Institute of Mathematical Statistics
  Abbreviation : Ann Appl Stat
Source Genre: Journal
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Publ. Info: Cleveland, OH, USA : Institute of Mathematical Statistics
Pages: - Volume / Issue: 3 (4) Sequence Number: - Start / End Page: 1285 - 1294 Identifier: ISSN: 1932-6157
CoNE: https://pure.mpg.de/cone/journals/resource/1932-6157