{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T19:21:54Z","timestamp":1791919314028,"version":"4.3.4"},"reference-count":42,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2017,5,21]],"date-time":"2017-05-21T00:00:00Z","timestamp":1495324800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100022491","name":"Max Planck ETH Center for Learning Systems","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100022491","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>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\u2013Schmidt 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\u2013Schmidt 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.<\/jats:p>","DOI":"10.1111\/rssb.12235","type":"journal-article","created":{"date-parts":[[2017,5,23]],"date-time":"2017-05-23T18:54:04Z","timestamp":1495565644000},"page":"5-31","source":"Crossref","is-referenced-by-count":139,"title":["Kernel-Based Tests for Joint Independence"],"prefix":"10.1093","volume":"80","author":[{"given":"Niklas","family":"Pfister","sequence":"first","affiliation":[{"name":"Eidgen\u00f6ssiche Technische Hochschule , Z\u00fcrich , Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"B\u00fchlmann","sequence":"additional","affiliation":[{"name":"Eidgen\u00f6ssiche Technische Hochschule , Z\u00fcrich , Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bernhard","family":"Sch\u00f6lkopf","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Intelligent Systems , T\u00fcbingen , Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonas","family":"Peters","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2017,5,21]]},"reference":[{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1016\/j.jmva.2005.10.005","article-title":"A multivariate nonparametric test of independence","volume":"97","author":"Bakirov","year":"2006","journal-title":"J. Multiv. Anal."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1214\/aos\/1176350497","article-title":"Stochastic estimation and testing","volume":"15","author":"Beran","year":"1987","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.3150\/13-BEJ514","article-title":"A consistent test of independence based on a sign covariance related to Kendall\u2019s tau","volume":"20","author":"Bergsma","year":"2014","journal-title":"Bernoulli"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2526","DOI":"10.1214\/14-AOS1260","article-title":"CAM: causal additive models, high-dimensional order search and penalized regression","volume":"42","author":"B\u00fchlmann","year":"2014","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2825","DOI":"10.1214\/009053606000000939","article-title":"Efficient independent component analysis","volume":"34","author":"Chen","year":"2006","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","first-page":"3608","volume-title":"Advances in Neural Information Processing Systems","author":"Chwialkowski","year":"2014"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511802843","volume-title":"Bootstrap Methods and Their Application","author":"Davison","year":"1997"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"419","DOI":"10.2307\/1403753","article-title":"A consistent test for bivariate dependence","volume":"61","author":"Feuerverger","year":"1993","journal-title":"Int. Statist. Rev."},{"key":"2023022005313796600_","first-page":"489","volume-title":"Advances in Neural Information Processing Systems","author":"Fukumizu","year":"2007"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2571","DOI":"10.1016\/j.jmva.2010.07.006","article-title":"A multivariate version of Hoeffding\u2019s phi-square","volume":"101","author":"Gai\u00dfer","year":"2010","journal-title":"J. Multiv. Anal."},{"key":"2023022005313796600_","first-page":"723","article-title":"A kernel two-sample test","volume":"13","author":"Gretton","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/11564089_7","volume-title":"Algorithmic Learning Theory","author":"Gretton","year":"2005"},{"key":"2023022005313796600_","first-page":"673","volume-title":"Advances in Neural Information Processing Systems","author":"Gretton","year":"2009"},{"key":"2023022005313796600_","first-page":"585","volume-title":"Advances in Neural Information Processing Systems","author":"Gretton","year":"2007"},{"key":"2023022005313796600_","first-page":"1205","volume-title":"Advances in Neural Information Processing Systems","author":"Gretton","year":"2012"},{"key":"2023022005313796600_","volume-title":"Consistent Testing of Total Independence based on the Empirical Characteristic Function","author":"Kankainen","year":"1995"},{"key":"2023022005313796600_","volume-title":"Testing Statistical Hypotheses","author":"Lehmann","year":"2005"},{"key":"2023022005313796600_","volume-title":"Testing independence in high dimensions with sums of squares of rank correlations","author":"Leung","year":"2016"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2293","DOI":"10.1214\/12-AOS1037","article-title":"High-dimensional semi-parametric gaussian copula graphical models","volume":"40","author":"Liu","year":"2012","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","article-title":"Independent component analysis via distance covariance","author":"Matteson","year":"2016","journal-title":"J. Am. Statist. Ass."},{"key":"2023022005313796600_","first-page":"1","article-title":"Distinguishing cause from effect using observational data: methods and benchmarks","volume":"17","author":"Mooij","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2287","DOI":"10.1214\/16-EJS1166","article-title":"Large-sample theory for the Bergsma-Dassios sign covariance","volume":"10","author":"Nandy","year":"2016","journal-title":"Electron. J. Statist."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511803161","volume-title":"Causality: Models, Reasoning, and Inference","author":"Pearl","year":"2009"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1162\/NECO_a_00708","article-title":"Structural intervention distance (SID) for evaluating causal graphs","volume":"27","author":"Peters","year":"2015","journal-title":"Neurl Computn"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2436","DOI":"10.1109\/TPAMI.2011.71","article-title":"Causal inference on discrete data using additive noise models","volume":"33","author":"Peters","year":"2011","journal-title":"IEEE Trans. Pattn Anal. Mach. Intell."},{"key":"2023022005313796600_","first-page":"2009","article-title":"Causal discovery with continuous additive noise models","volume":"15","author":"Peters","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"2023022005313796600_","volume-title":"Joint independence testing. Master\u2019s Thesis.","author":"Pfister","year":"2016"},{"key":"2023022005313796600_","volume-title":"A bootstrap revival of some nonparametric tests","author":"Romano","year":"1986"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1080\/01621459.1988.10478650","article-title":"A bootstrap revival of some nonparametric distance tests","volume":"83","author":"Romano","year":"1988","journal-title":"J. Am. Statist. Ass."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1214\/aos\/1176347007","article-title":"Bootstrap and randomization tests of some nonparametric hypotheses","volume":"17","author":"Romano","year":"1989","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"110","DOI":"10.2307\/3002019","article-title":"An approximate distribution of estimates of variance components","volume":"2","author":"Satterthwaite","year":"1946","journal-title":"Biometr. Bull."},{"key":"2023022005313796600_","first-page":"1124","volume-title":"Advances in Neural Information Processing Systems","author":"Sejdinovic","year":"2013"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316481","volume-title":"Approximation Theorems of Mathematical Statistics","author":"Serfling","year":"1980"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/978-3-540-75225-7_5","volume-title":"Algorithmic Learning Theory","author":"Smola","year":"2007"},{"key":"2023022005313796600_","volume-title":"Proc. Conf. Learning Theory","author":"Sriperumbudur","year":"2008"},{"key":"2023022005313796600_","first-page":"1236","article-title":"Brownian distance covariance","volume":"3","author":"Sz\u00e9kely","year":"2009","journal-title":"Ann. Appl. Statist."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2382","DOI":"10.1214\/14-AOS1255","article-title":"Partial distance correlation with methods for dissimilarities","volume":"42","author":"Sz\u00e9kely","year":"2014","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107415805","volume-title":"An Introduction to Sparse Stochastic Processes","author":"Unser","year":"2014"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.3150\/14-BEJ690","article-title":"Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas","volume":"22","author":"Wegkamp","year":"2016","journal-title":"Bernoulli"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0304-3800(02)00193-X","article-title":"GAMs with integrated model selection using penalized regression splines and applications to environmental modelling","volume":"157","author":"Wood","year":"2002","journal-title":"Ecol. Modllng"},{"key":"2023022005313796600_","doi-asserted-by":"crossref","first-page":"2541","DOI":"10.1214\/12-AOS1041","article-title":"Regularized rank-based estimation of high-dimensional non-paranormal graphical models","volume":"40","author":"Xue","year":"2012","journal-title":"Ann. Statist."},{"key":"2023022005313796600_","first-page":"1","article-title":"Large-scale kernel methods for independence testing","volume":"27","author":"Zhang","year":"2017","journal-title":"Statist. Comput."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12235","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/80\/1\/5\/49271330\/jrsssb_80_1_5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/80\/1\/5\/49271330\/jrsssb_80_1_5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,24]],"date-time":"2024-06-24T04:54:33Z","timestamp":1719204873000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/80\/1\/5\/7048410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,21]]},"references-count":42,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,5,21]]},"published-print":{"date-parts":[[2018,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12235","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2018,1]]},"published":{"date-parts":[[2017,5,21]]}}}