{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T03:40:03Z","timestamp":1785210003300,"version":"3.55.0"},"reference-count":28,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2025,7,21]],"date-time":"2025-07-21T00:00:00Z","timestamp":1753056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,2,13]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Common workflows in machine learning and statistics rely on the ability to partition the information in a dataset into independent portions. Recent work has shown that this may be possible even when conventional sample splitting is not, such as when the number of samples, $ n $, is one or when observations are not independent and identically distributed. In the case of multivariate Gaussian data, these alternatives to sample splitting require knowledge of the covariance matrix. In many important problems, such as in spatial or longitudinal data analysis and in graphical modelling, the covariance matrix may be unknown and even of primary interest. Therefore, in this work we develop new approaches for decomposing multivariate Gaussians with unknown covariance. First, we present a general algorithm that encompasses all previous decomposition methods for Gaussian data as special cases and which can further handle the case of unknown covariance. It yields a new and more flexible alternative to sample splitting when $ n \\,{\\gt}\\, 1 $. When $ n=1 $, we prove that it is impossible to partition the information in a multivariate Gaussian into independent portions without knowing the covariance matrix. Hence, we use the general algorithm to decompose a single multivariate Gaussian with unknown covariance into dependent parts with tractable conditional distributions and demonstrate their use for inference and validation. The proposed decomposition strategy extends naturally to Gaussian processes. In simulations and for electroencephalography data, we apply these decompositions to the tasks of model selection and post-selection inference in settings where alternative strategies are unavailable.<\/jats:p>","DOI":"10.1093\/biomet\/asaf057","type":"journal-article","created":{"date-parts":[[2025,7,17]],"date-time":"2025-07-17T10:11:51Z","timestamp":1752747111000},"source":"Crossref","is-referenced-by-count":4,"title":["Decomposing Gaussians with unknown covariance"],"prefix":"10.1093","volume":"113","author":[{"given":"A","family":"Dharamshi","sequence":"first","affiliation":[{"name":"Department of Biostatistics, University of Washington , Seattle, Washington 98195,","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A","family":"Neufeld","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Williams College , 18 Hoxsey Street , Williamstown, Massachusetts 01267, . acn2@williams.edu","place":["U.S.A"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L L","family":"Gao","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of British Columbia , Vancouver, British Columbia V6T 1Z4, , lucy.gao@ubc.ca","place":["Canada"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J","family":"Bien","sequence":"additional","affiliation":[{"name":"Department of Data Science and Operations, University of Southern California , 3670 Trousdale Parkway , Los Angeles, California 90007, \u00a0 jbien@usc.edu","place":["U.S.A."]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D","family":"Witten","sequence":"additional","affiliation":[{"name":"Departments of Biostatistics and Statistics, University of Washington , Seattle, Washington 98195, \u00a0 dwitten@uw.edu","place":["U.S.A."]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,7,21]]},"reference":[{"key":"2026032607093517200_asaf057-B1","doi-asserted-by":"publisher","DOI":"10.24432\/C5TS3D","volume-title":"EEG Database","author":"Begleiter","year":"1999"},{"key":"2026032607093517200_asaf057-B2","doi-asserted-by":"crossref","first-page":"2499","DOI":"10.1080\/00949655.2020.1783262","article-title":"Approximate leave-future-out cross-validation for Bayesian time series models","volume":"90","author":"B\u00fcrkner","year":"2020","journal-title":"J. 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