{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T03:06:26Z","timestamp":1773543986242,"version":"3.50.1"},"reference-count":13,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,3]],"date-time":"2017-07-03T00:00:00Z","timestamp":1499040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the problem of quantifying the information shared by a pair of random variables      X 1  ,  X 2      about another variable S. We propose a new measure of shared information, called extractable shared information, that is left monotonic; that is, the information shared about S is bounded from below by the information shared about     f ( S )     for any function f. We show that our measure leads to a new nonnegative decomposition of the mutual information     I ( S ;  X 1   X 2  )     into shared, complementary and unique components. We study properties of this decomposition and show that a left monotonic shared information is not compatible with a Blackwell interpretation of unique information. We also discuss whether it is possible to have a decomposition in which both shared and unique information are left monotonic.<\/jats:p>","DOI":"10.3390\/e19070328","type":"journal-article","created":{"date-parts":[[2017,7,3]],"date-time":"2017-07-03T10:27:31Z","timestamp":1499077651000},"page":"328","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["On Extractable Shared Information"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6411-7775","authenticated-orcid":false,"given":"Johannes","family":"Rauh","sequence":"first","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"given":"Pradeep","family":"Banerjee","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3987-1271","authenticated-orcid":false,"given":"Eckehard","family":"Olbrich","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5258-6590","authenticated-orcid":false,"given":"J\u00fcrgen","family":"Jost","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"given":"Nils","family":"Bertschinger","sequence":"additional","affiliation":[{"name":"Frankfurt Institute for Advanced Studies, 60438 Frankfurt, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2161","DOI":"10.3390\/e16042161","article-title":"Quantifying unique information","volume":"16","author":"Bertschinger","year":"2014","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Rauh, J., Bertschinger, N., Olbrich, E., and Jost, J. (July, January 29). Reconsidering unique information: Towards a multivariate information decomposition. Proceedings of 2014 IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, USA.","DOI":"10.1109\/ISIT.2014.6875230"},{"key":"ref_3","unstructured":"Williams, P., and Beer, R. (arXiv, 2010). Nonnegative Decomposition of Multivariate Information, arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"012130","DOI":"10.1103\/PhysRevE.87.012130","article-title":"A Bivariate measure of redundant information","volume":"87","author":"Harder","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bertschinger, N., Rauh, J., Olbrich, E., and Jost, J. (2012, January 2\u20137). Shared Information\u2014New Insights and Problems in Decomposing Information in Complex Systems. Proceedings of the European Conference on Complex Systems 2012, Brussels, Belgium.","DOI":"10.1007\/978-3-319-00395-5_35"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/978-3-642-53734-9_6","article-title":"Quantifying Synergistic Mutual Information","volume":"Volume 9","author":"Prokopenko","year":"2014","journal-title":"Guided Self-Organization: Inception"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.bandc.2015.09.004","article-title":"Partial information decomposition as a unified approach to the specification of neural goal functions","volume":"112","author":"Wibral","year":"2017","journal-title":"Brain Cogn."},{"key":"ref_8","unstructured":"Bell, A.J. (2003, January 1\u20134). The Co-Information Lattice. 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Proceedings of 1997 IEEE International Symposium on Information Theory, Ulm, Germany."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.3390\/e16041985","article-title":"Intersection Information Based on Common Randomness","volume":"16","author":"Griffith","year":"2014","journal-title":"Entropy"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rauh, J., Banerjee, P.K., Olbrich, E., Jost, J., Bertschinger, N., and Wolpert, D. (arXiv, 2017). 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