{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T04:48:46Z","timestamp":1778734126055,"version":"3.51.4"},"reference-count":23,"publisher":"Oxford University Press (OUP)","license":[{"start":{"date-parts":[[2019,10,25]],"date-time":"2019-10-25T00:00:00Z","timestamp":1571961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We consider graphical models based on a recursive system of linear structural equations. This implies that there is an ordering, $\\sigma$, of the variables such that each observed variable $Y_v$ is a linear function of a variable-specific error term and the other observed variables $Y_u$ with $\\sigma(u) &lt; \\sigma (v)$. The causal relationships, i.e., which other variables the linear functions depend on, can be described using a directed graph. It has previously been shown that when the variable-specific error terms are non-Gaussian, the exact causal graph, as opposed to a Markov equivalence class, can be consistently estimated from observational data. We propose an algorithm that yields consistent estimates of the graph also in high-dimensional settings in which the number of variables may grow at a faster rate than the number of observations, but in which the underlying causal structure features suitable sparsity; specifically, the maximum in-degree of the graph is controlled. Our theoretical analysis is couched in the setting of log-concave error distributions.<\/jats:p>","DOI":"10.1093\/biomet\/asz055","type":"journal-article","created":{"date-parts":[[2019,9,12]],"date-time":"2019-09-12T19:31:57Z","timestamp":1568316717000},"source":"Crossref","is-referenced-by-count":7,"title":["High-dimensional causal discovery under non-Gaussianity"],"prefix":"10.1093","author":[{"given":"Y Samuel","family":"Wang","sequence":"first","affiliation":[{"name":"Booth School of Business, The University of Chicago, 5807 South Woodlawn Avenue, Chicago, Illinois, U.S.A"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mathias","family":"Drton","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Technical University of Munich, Boltzmannstra\u00dfe 3, Garching bei M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2019,10,25]]},"reference":[{"key":"2019102505494133000_B1","first-page":"3741","article-title":"Order-independent constraint-based causal structure learning","volume":"15","author":"Colombo,","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B2","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1146\/annurev-statistics-060116-053803","article-title":"Structure learning in graphical modeling","volume":"4","author":"Drton,","year":"2017","journal-title":"Ann. Rev. Statist. Appl."},{"key":"2019102505494133000_B3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v040.i08","article-title":"Rcpp: Seamless R and C++ integration","volume":"40","author":"Eddelbuettel,","year":"2011","journal-title":"J. Statist. Software"},{"key":"2019102505494133000_B4","doi-asserted-by":"crossref","DOI":"10.1186\/1752-0509-6-34","article-title":"Revisiting the variation of clustering coefficient of biological networks suggests new modular structure","volume":"6","author":"Hao,","year":"2012","journal-title":"BMC Syst. Biol."},{"key":"2019102505494133000_B5","first-page":"3365","article-title":"PC algorithm for nonparanormal graphical models","volume":"14","author":"Harris,","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B6","first-page":"282","article-title":"Causal discovery of linear acyclic models with arbitrary distributions","volume-title":"Proc. 24th Conf. Uncert. Artif. Intel. (UAI 2008, Helsinki, Finland, July 9\u201312)","author":"Hoyer,","year":"2008"},{"key":"2019102505494133000_B7","first-page":"111","article-title":"Pairwise likelihood ratios for estimation of non-Gaussian structural equation models","volume":"14","author":"Hyv\u00e4rinen,","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B8","first-page":"613","article-title":"Estimating high-dimensional directed acyclic graphs with the PC-algorithm","volume":"8","author":"Kalisch,","year":"2007","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B9","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1214\/16-EJS1126","article-title":"Estimation of high-dimensional graphical models using regularized score matching","volume":"10","author":"Lin,","year":"2016","journal-title":"Electron. J. Statist."},{"key":"2019102505494133000_B10","first-page":"3065","article-title":"High-dimensional learning of linear causal networks via inverse covariance estimation","volume":"15","author":"Loh,","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B11","doi-asserted-by":"crossref","first-page":"3133","DOI":"10.1214\/09-AOS685","article-title":"Estimating high-dimensional intervention effects from observational data","volume":"37","author":"Maathuis,","year":"2009","journal-title":"Ann. Statist."},{"key":"2019102505494133000_B12","doi-asserted-by":"crossref","first-page":"1436","DOI":"10.1214\/009053606000000281","article-title":"High-dimensional graphs and variable selection with the lasso","volume":"34","author":"Meinshausen,","year":"2006","journal-title":"Ann. Statist."},{"key":"2019102505494133000_B13","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1214\/aos\/1176342472","article-title":"Distinctness of the eigenvalues of a quadratic form in a multivariate sample","volume":"1","author":"Okamoto,","year":"1973","journal-title":"Ann. Statist."},{"key":"2019102505494133000_B14","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511803161","volume-title":"Causality: Models, Reasoning, and Inference","author":"Pearl,","year":"2009","edition":"2nd ed"},{"key":"2019102505494133000_B15","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1093\/biomet\/ast043","article-title":"Identifiability of Gaussian structural equation models with equal error variances","volume":"101","author":"Peters,","year":"2014","journal-title":"Biometrika"},{"key":"2019102505494133000_B16","year":"2019","journal-title":"R: A Language and Environment for Statistical Computing"},{"key":"2019102505494133000_B17","doi-asserted-by":"crossref","first-page":"2904","DOI":"10.1214\/17-AOS1643","article-title":"Causal inference in partially linear structural equation models","volume":"46","author":"Rothenh\u00e4usler,","year":"2018","journal-title":"Ann. Statist."},{"key":"2019102505494133000_B18","first-page":"2003","article-title":"A linear non-Gaussian acyclic model for causal discovery","volume":"7","author":"Shimizu,","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B19","first-page":"1225","article-title":"DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model","volume":"12","author":"Shimizu,","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"2019102505494133000_B20","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1214\/14-EJS932","article-title":"Quantifying identifiability in independent component analysis","volume":"8","author":"Sokol,","year":"2014","journal-title":"Electron. J. Statist."},{"key":"2019102505494133000_B21","volume-title":"Causation, Prediction, and Search","author":"Spirtes,","year":"2000","edition":"2nd ed"},{"key":"2019102505494133000_B22","first-page":"485","article-title":"Enumeration of labelled essential graphs","volume":"111","author":"Steinsky,","year":"2013","journal-title":"Ars Combin."},{"key":"2019102505494133000_B23","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1214\/12-AOS1080","article-title":"Geometry of the faithfulness assumption in causal inference","volume":"41","author":"Uhler,","year":"2013","journal-title":"Ann. Statist."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asz055\/30300595\/asz055.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asz055\/30300595\/asz055.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,25]],"date-time":"2019-10-25T09:52:16Z","timestamp":1571997136000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article\/doi\/10.1093\/biomet\/asz055\/5607065"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,25]]},"references-count":23,"URL":"https:\/\/doi.org\/10.1093\/biomet\/asz055","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,25]]},"article-number":"asz055"}}