{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T17:07:54Z","timestamp":1780765674785,"version":"3.54.1"},"reference-count":17,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5915,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1990,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate directed Markov fields over finite graphs without positivity assumptions on the densities involved. A criterion for conditional independence of two groups of variables given a third is given and named as the directed, global Markov property. We give a simple proof of the fact that the directed, local Markov property and directed, global Markov property are equivalent and \u2013 in the case of absolute continuity w. r. t. a product measure \u2013 equivalent to the recursive factorization of densities. It is argued that our criterion is easy to use, it is sharper than that given by Kiiveri, Speed, and Carlin and equivalent to that of Pearl. It follows that our criterion cannot be sharpened.<\/jats:p>","DOI":"10.1002\/net.3230200503","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T09:22:22Z","timestamp":1178961742000},"page":"491-505","source":"Crossref","is-referenced-by-count":325,"title":["Independence properties of directed markov fields"],"prefix":"10.1002","volume":"20","author":[{"given":"S. L.","family":"Lauritzen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A. 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Stat. \u00c5arhus University \u00c5arhus Denmark (1989)."},{"key":"e_1_2_1_5_2","article-title":"Marginalization and collapsibility in graphical association models","author":"Frydenberg M.","journal-title":"Ann. Statist"},{"key":"e_1_2_1_6_2","unstructured":"D.GeigerandJ.Pearl On the logic of influence diagrams. Proc. 4th Workshop on Uncertainty in Artificial Intelligence Minneapolis MN (1988)136\u2013147."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700027312"},{"key":"e_1_2_1_8_2","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1111\/j.2517-6161.1988.tb01721.x","article-title":"Local computations with probabilities on graphical structures and their application to expert systems (with discussion)","volume":"50","author":"Lauritzen S. L.","year":"1988","journal-title":"J. R. Statist. Soc. 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