{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T16:19:51Z","timestamp":1790007591722,"version":"4.0.1"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,26]],"date-time":"2023-02-26T00:00:00Z","timestamp":1677369600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["JPMJSP2138"],"award-info":[{"award-number":["JPMJSP2138"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["18K11192"],"award-info":[{"award-number":["18K11192"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Grant-in-Aid for Scientific Research (C)","award":["JPMJSP2138"],"award-info":[{"award-number":["JPMJSP2138"]}]},{"name":"Grant-in-Aid for Scientific Research (C)","award":["18K11192"],"award-info":[{"award-number":["18K11192"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Conditional Independence (CI) test is a fundamental problem in statistics. Many nonparametric CI tests have been developed, but a common challenge exists: the current methods perform poorly with a high-dimensional conditioning set. In this paper, we considered a nonparametric CI test using a kernel-based test statistic, which can be viewed as an extension of the Hilbert\u2013Schmidt Independence Criterion (HSIC). We propose a local bootstrap method to generate samples from the null distribution H0:X\u2aebY\u2223Z. The experimental results showed that our proposed method led to a significant performance improvement compared with previous methods. In particular, our method performed well against the growth of the dimension of the conditioning set. Meanwhile, our method can be computed efficiently against the growth of the sample size and the dimension of the conditioning set.<\/jats:p>","DOI":"10.3390\/e25030425","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T03:23:36Z","timestamp":1677468216000},"page":"425","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Extending Hilbert\u2013Schmidt Independence Criterion for Testing Conditional Independence"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4892-323X","authenticated-orcid":false,"given":"Bingyuan","family":"Zhang","sequence":"first","affiliation":[{"name":"Graduate School of Engineer Science, Osaka University, Toyonaka 560-0043, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3195-9922","authenticated-orcid":false,"given":"Joe","family":"Suzuki","sequence":"additional","affiliation":[{"name":"Graduate School of Engineer Science, Osaka University, Toyonaka 560-0043, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,26]]},"reference":[{"key":"ref_1","unstructured":"Pearl, J. 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