{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T02:58:02Z","timestamp":1776135482246,"version":"3.50.1"},"reference-count":30,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2009,5,27]],"date-time":"2009-05-27T00:00:00Z","timestamp":1243382400000},"content-version":"vor","delay-in-days":146,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/D03261X\/1"],"award-info":[{"award-number":["EP\/D03261X\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["107E021"],"award-info":[{"award-number":["107E021"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["09A105P"],"award-info":[{"award-number":["09A105P"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004410","name":"Scientific and Technological Research Council of Turkey","doi-asserted-by":"crossref","award":["EP\/D03261X\/1"],"award-info":[{"award-number":["EP\/D03261X\/1"]}],"id":[{"id":"10.13039\/501100004410","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004410","name":"Scientific and Technological Research Council of Turkey","doi-asserted-by":"crossref","award":["107E021"],"award-info":[{"award-number":["107E021"]}],"id":[{"id":"10.13039\/501100004410","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004410","name":"Scientific and Technological Research Council of Turkey","doi-asserted-by":"crossref","award":["09A105P"],"award-info":[{"award-number":["09A105P"]}],"id":[{"id":"10.13039\/501100004410","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100005037","name":"Bo\u011fazi\u00e7i University","doi-asserted-by":"crossref","award":["EP\/D03261X\/1"],"award-info":[{"award-number":["EP\/D03261X\/1"]}],"id":[{"id":"10.13039\/501100005037","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100005037","name":"Bo\u011fazi\u00e7i University","doi-asserted-by":"crossref","award":["107E021"],"award-info":[{"award-number":["107E021"]}],"id":[{"id":"10.13039\/501100005037","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100005037","name":"Bo\u011fazi\u00e7i University","doi-asserted-by":"crossref","award":["09A105P"],"award-info":[{"award-number":["09A105P"]}],"id":[{"id":"10.13039\/501100005037","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Computational Intelligence and Neuroscience"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>We describe nonnegative matrix factorisation (NMF) with a Kullback\u2010Leibler (KL) error measure in a statistical framework, with a hierarchical generative model consisting of an observation and a prior component. Omitting the prior leads to the standard KL\u2010NMF algorithms as special cases, where maximum likelihood parameter estimation is carried out via the Expectation\u2010Maximisation (EM) algorithm. Starting from this view, we develop full Bayesian inference via variational Bayes or Monte Carlo. Our construction retains conjugacy and enables us to develop more powerful models while retaining attractive features of standard NMF such as monotonic convergence and easy implementation. We illustrate our approach on model order selection and image reconstruction.<\/jats:p>","DOI":"10.1155\/2009\/785152","type":"journal-article","created":{"date-parts":[[2009,5,27]],"date-time":"2009-05-27T13:46:39Z","timestamp":1243431999000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":212,"title":["Bayesian Inference for Nonnegative Matrix Factorisation Models"],"prefix":"10.1155","volume":"2009","author":[{"given":"Ali Taylan","family":"Cemgil","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2009,5,27]]},"reference":[{"key":"e_1_2_17_1_2","doi-asserted-by":"publisher","DOI":"10.1038\/44565"},{"key":"e_1_2_17_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/env.3170050203"},{"key":"e_1_2_17_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-45528-0"},{"key":"e_1_2_17_4_2","unstructured":"VirtanenT. Sound source separation in monaural music signals Ph.D. thesis 2006 NovemberTampere University of Technology Tampere Finland."},{"key":"e_1_2_17_5_2","unstructured":"KameokaH. Statistical approach to multipitch analysis Ph.D. thesis 2007 University of Tokyo Tokyo Japan."},{"key":"e_1_2_17_6_2","doi-asserted-by":"publisher","DOI":"10.1155\/2008\/852187"},{"key":"e_1_2_17_7_2","volume-title":"Poisson Processes","author":"Kingman J. F. C.","year":"1993"},{"key":"e_1_2_17_8_2","unstructured":"LeeD. D.andSeungH. S. Algorithms for non-negative matrix factorization 13 Proceedings of the Conference on Advances in Neural Information Processing Systems (NIPS \u203200) October-November 2000 Denver Colo USA 556\u2013562."},{"key":"e_1_2_17_9_2","unstructured":"Le RouxJ. KameokaH. OnoN. andSagayamaS. On the relation between divergence-based minimization and maximum-likelihood estimation for the i-divergence Personal Communication 2008."},{"key":"e_1_2_17_10_2","doi-asserted-by":"crossref","unstructured":"VirtanenT. CemgilA. T. andGodsillS. Bayesian extensions to non-negative matrix factorisation for audio signal modelling Proceedings of IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP \u203208) March-April 2008 Las Vegas Nev USA 1825\u20131828 https:\/\/doi.org\/10.1109\/ICASSP.2008.4517987.","DOI":"10.1109\/ICASSP.2008.4517987"},{"key":"e_1_2_17_11_2","doi-asserted-by":"crossref","unstructured":"GaussierE.andGoutteC. Relation between PLSA and NMF and implication Proceedings of the 28th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval August 2005 Salvador Brazil 601\u2013602.","DOI":"10.1145\/1076034.1076148"},{"key":"e_1_2_17_12_2","doi-asserted-by":"crossref","unstructured":"SinghA. P.andajit@cs.cmu.edu GordonG. J. ggordon@cs.cmu.edu A unified view of matrix factorization models 5212 Proceedings of the European Conference on Machine Learning and Knowledge Discovery in Databases (ECML PKDD \u203208) September 2008 Antwerp Belgium Springer 358\u2013373 Lecture Notes in Computer Science https:\/\/doi.org\/10.1007\/978-3-540-87481-2_24.","DOI":"10.1007\/978-3-540-87481-2_24"},{"key":"e_1_2_17_13_2","doi-asserted-by":"publisher","DOI":"10.1155\/2008\/361705"},{"key":"e_1_2_17_14_2","first-page":"507","volume-title":"Advances in Neural Information Processing Systems","author":"Ghahramani Z.","year":"2000"},{"key":"e_1_2_17_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-4485-6"},{"key":"e_1_2_17_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-76371-2"},{"key":"e_1_2_17_17_2","first-page":"1313","article-title":"Marginal likelihood from the gibbs output","volume":"90","author":"Chib S.","year":"1995","journal-title":"Journal of the Acoustical Society of America"},{"key":"e_1_2_17_18_2","first-page":"1457","article-title":"Non-negative matrix factorization with sparseness constraints","volume":"5","author":"Hoyer P. O.","year":"2004","journal-title":"The Journal of Machine Learning Research"},{"key":"e_1_2_17_19_2","first-page":"1705","article-title":"Clustering with Bregman divergences","volume":"6","author":"Banerjee A.","year":"2005","journal-title":"Journal of Machine Learning Research"},{"key":"e_1_2_17_20_2","unstructured":"ShashankaM. V. RajB. andSmaragdisP. Sparse overcomplete latent variable decomposition of counts data Proceedings of the Conference on Advances in Neural Information Processing Systems (NIPS \u203207) December 2007 Vancouver Canada."},{"key":"e_1_2_17_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-9473(93)90115-A"},{"key":"e_1_2_17_22_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4024-2"},{"key":"e_1_2_17_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-9473(98)00075-9"},{"key":"e_1_2_17_24_2","doi-asserted-by":"crossref","unstructured":"ButlerK.andStephensM. The distribution of a sum of binomial random variables 1993 Aprilno. 467 Stanford University Stanford Calif USA prepared for the Office of Naval Research.","DOI":"10.21236\/ADA266969"},{"key":"e_1_2_17_25_2","unstructured":"MinkaT. Expectation propagation for approximate Bayesian inference Ph.D. thesis 2001 MIT Media Lab Cambridge Mass USA."},{"key":"e_1_2_17_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dsp.2007.01.003"},{"key":"e_1_2_17_27_2","doi-asserted-by":"crossref","unstructured":"BuntineW. Variational extensions to EM and multinomial PCA 2430 Proceedings of the 13th European Conference on Machine Learning (ECML \u203202) August 2002 Helsinki Finland Springer 23\u201334 Lecture Notes In Computer Science.","DOI":"10.1007\/3-540-36755-1_3"},{"key":"e_1_2_17_28_2","doi-asserted-by":"crossref","unstructured":"CemgilA. T. atc27@eng.cam.ac.uk PeelingP. php23@eng.cam.ac.uk DikmenO. andod225@eng.cam.ac.uk GodsillS. sjg@eng.cam.ac.uk Prior structures for time-frequency energy distributions Proceedings of IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA \u203207) October 2007 New Paltz NY USA 151\u2013154 https:\/\/doi.org\/10.1109\/ASPAA.2007.4393007.","DOI":"10.1109\/ASPAA.2007.4393007"},{"key":"e_1_2_17_29_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00220-5"},{"key":"e_1_2_17_30_2","doi-asserted-by":"publisher","DOI":"10.1109\/18.761251"}],"container-title":["Computational Intelligence and Neuroscience"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/cin\/2009\/785152.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/cin\/2009\/785152.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2009\/785152","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,9]],"date-time":"2025-02-09T18:32:35Z","timestamp":1739125955000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2009\/785152"}},"subtitle":[],"editor":[{"given":"S.","family":"Cruces-Alvarez","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1155\/2009\/785152"],"URL":"https:\/\/doi.org\/10.1155\/2009\/785152","archive":["Portico"],"relation":{},"ISSN":["1687-5265","1687-5273"],"issn-type":[{"value":"1687-5265","type":"print"},{"value":"1687-5273","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]},"assertion":[{"value":"2008-08-29","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-02-14","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-05-27","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"785152"}}