{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T06:36:59Z","timestamp":1786171019738,"version":"3.56.0"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2014,4,15]],"date-time":"2014-04-15T00:00:00Z","timestamp":1397520000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We propose new measures of shared information, unique information and synergistic information that can be used to decompose the mutual information of a pair of random variables (Y, Z) with a third random variable X. Our measures are motivated by an operational idea of unique information, which suggests that shared information and unique information should depend only on the marginal distributions of the pairs (X, Y) and (X,Z). Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.<\/jats:p>","DOI":"10.3390\/e16042161","type":"journal-article","created":{"date-parts":[[2014,4,15]],"date-time":"2014-04-15T13:23:16Z","timestamp":1397568196000},"page":"2161-2183","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":238,"title":["Quantifying Unique Information"],"prefix":"10.3390","volume":"16","author":[{"given":"Nils","family":"Bertschinger","sequence":"first","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 23, 04109 Leipzig, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6411-7775","authenticated-orcid":false,"given":"Johannes","family":"Rauh","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 23, 04109 Leipzig, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eckehard","family":"Olbrich","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 23, 04109 Leipzig, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00fcrgen","family":"Jost","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 23, 04109 Leipzig, Germany"},{"name":"Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM 87501, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nihat","family":"Ay","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 23, 04109 Leipzig, Germany"},{"name":"Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM 87501, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2014,4,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF02289159","article-title":"Multivariate information transmission","volume":"19","author":"McGill","year":"1954","journal-title":"Psychometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"11539","DOI":"10.1523\/JNEUROSCI.23-37-11539.2003","article-title":"Synergy, redundancy, and independence in population codes","volume":"23","author":"Schneidman","year":"2003","journal-title":"J. Neurosci"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5195","DOI":"10.1523\/JNEUROSCI.5319-04.2005","article-title":"Synergy, redundancy, and independence in population codes, revisited","volume":"25","author":"Latham","year":"2005","journal-title":"J. Neurosci"},{"key":"ref_4","unstructured":"Williams, P., and Beer, R. (2010). Nonnegative decomposition of multivariate information. arXiv:1004.2515v1."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Griffith, V., and Koch, C. (2013). Quantifying synergistic mutual information. arXiv:1205.4265.","DOI":"10.1007\/978-3-642-53734-9_6"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"012130","DOI":"10.1103\/PhysRevE.87.012130","article-title":"Bivariate measure of redundant information","volume":"87","author":"Harder","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bertschinger, N., Rauh, J., Olbrich, E., and Jost, J. (2012, January September). Shared information\u2013new insights and problems in decomposing Information in Complex Systems. Brussels.","DOI":"10.1007\/978-3-319-00395-5_35"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bertschinger, N., and Rauh, J. (2014). The blackwell relation defines no lattice. arXiv:1401.3146.","DOI":"10.1109\/ISIT.2014.6875280"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1214\/aoms\/1177729032","article-title":"Equivalent comparisons of experiments","volume":"24","author":"Blackwell","year":"1953","journal-title":"Ann. Math. Stat"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Rauh, J., Bertschinger, N., Olbrich, E., and Jost, J. (2014). Reconsidering unique information: Towards a multivariate information Decomposition. Accepted for ISIT 2014.","DOI":"10.1109\/ISIT.2014.6875230"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1006\/jcta.2002.3301","article-title":"Gr\u00f6bner bases and polyhedral geometry of reducible and cyclic Models","volume":"100","author":"Sullivant","year":"2002","journal-title":"J. Combin. Theor. A"},{"key":"ref_12","unstructured":"(2003). Mathematica, Wolfram Research Inc. [8th ed]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/4\/2161\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:10:17Z","timestamp":1760217017000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/4\/2161"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,15]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,4]]}},"alternative-id":["e16042161"],"URL":"https:\/\/doi.org\/10.3390\/e16042161","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,4,15]]}}}