{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:32:14Z","timestamp":1760239934137,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,1,24]],"date-time":"2019-01-24T00:00:00Z","timestamp":1548288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I3073"],"award-info":[{"award-number":["I3073"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the world of generalized entropies\u2014which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom\u2014there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both appear naturally in different contexts. Linear constraints appear, e.g., in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here, we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of    \u03d5   -deformed exponential families that correspond to linear constraints (as studied by J.Naudts) to those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of     ( c , d )    -entropy, which covers all situations that are compatible with the first three Shannon\u2013Khinchin axioms and that include Shannon, Tsallis, Anteneodo\u2013Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality and mention that the escort distributions arising in these two dualities are generally different and only coincide for the case of the Tsallis deformation.<\/jats:p>","DOI":"10.3390\/e21020112","type":"journal-article","created":{"date-parts":[[2019,1,24]],"date-time":"2019-01-24T11:12:48Z","timestamp":1548328368000},"page":"112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Information Geometric Duality of \u03d5-Deformed Exponential Families"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5371-5320","authenticated-orcid":false,"given":"Jan","family":"Korbel","sequence":"first","affiliation":[{"name":"Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, A-1080 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rudolf","family":"Hanel","sequence":"additional","affiliation":[{"name":"Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, A-1080 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Thurner","sequence":"additional","affiliation":[{"name":"Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, A-1080 Vienna, Austria"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"},{"name":"IIASA, Schlossplatz 1, A-2361 Laxenburg, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"032124","DOI":"10.1103\/PhysRevE.96.032124","article-title":"Three faces of entropy for complex systems: Information, thermodynamics, and the maximum entropy principle","volume":"96","author":"Thurner","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"191","DOI":"10.3390\/e3030191","article-title":"Maximum entropy fundamentals","volume":"3","year":"2001","journal-title":"Entropy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"056125","DOI":"10.1103\/PhysRevE.66.056125","article-title":"Statistical mechanics in the context of special relativity","volume":"66","author":"Kaniadakis","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.aop.2004.01.002","article-title":"The world according to R\u00e9nyi: Thermodynamics of multifractal systems","volume":"312","author":"Jizba","year":"2004","journal-title":"Ann. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2487","DOI":"10.1140\/epjc\/s10052-013-2487-6","article-title":"Black hole thermodynamical entropy","volume":"73","author":"Tsallis","year":"2013","journal-title":"Eur. Phys. J. C"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Thurner, S., Hanel, R., and Klimek, P. (2018). Introduction to the Theory of Complex Systems, Oxford University Press.","DOI":"10.1093\/oso\/9780198821939.001.0001"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20006","DOI":"10.1209\/0295-5075\/93\/20006","article-title":"A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and distribution functions","volume":"93","author":"Hanel","year":"2011","journal-title":"Europhys. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"50003","DOI":"10.1209\/0295-5075\/96\/50003","article-title":"When do generalized entropies apply? How phase space volume determines entropy","volume":"96","author":"Hanel","year":"2011","journal-title":"Europhys. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"15377","DOI":"10.1073\/pnas.0503807102","article-title":"Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive","volume":"102","author":"Tsallis","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"375002","DOI":"10.1088\/1751-8121\/aad57b","article-title":"Statistical mechanics of exploding phase spaces: Ontic open systems","volume":"51","author":"Jensen","year":"2018","journal-title":"J. Phys. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"093007","DOI":"10.1088\/1367-2630\/aadcbe","article-title":"Classification of complex systems by their sample-space scaling exponents","volume":"20","author":"Korbel","year":"2018","journal-title":"New J. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/S0378-4371(98)00437-3","article-title":"The role of constraints within generalized nonextensive statistics","volume":"261","author":"Tsallis","year":"1998","journal-title":"Physica A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"031101","DOI":"10.1103\/PhysRevE.68.031101","article-title":"Geometry of escort distributions","volume":"68","author":"Abe","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"012012","DOI":"10.1088\/1742-6596\/201\/1\/012012","article-title":"A dually flat structure on the space of escort distributions","volume":"201","author":"Ohara","year":"2010","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4460","DOI":"10.1016\/j.physa.2012.04.024","article-title":"A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians","volume":"391","author":"Bercher","year":"2012","journal-title":"Physica A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"20005","DOI":"10.1209\/0295-5075\/85\/20005","article-title":"On the robustness of q-expectation values and Renyi entropy","volume":"85","author":"Hanel","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1140\/epjb\/e2009-00330-1","article-title":"Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example","volume":"72","author":"Hanel","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"026106","DOI":"10.1103\/PhysRevE.67.026106","article-title":"Constructing a statistical mechanics for Beck-Cohen superstatistics","volume":"67","author":"Tsallis","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Physica A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"19151","DOI":"10.1073\/pnas.1216885109","article-title":"Generalized entropies and logarithms and their duality relations","volume":"109","author":"Hanel","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_23","first-page":"183","article-title":"Information geometry of divergence functions","volume":"58","author":"Amari","year":"2010","journal-title":"Bull. Pol. Acad. Sci. Tech. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4308","DOI":"10.1016\/j.physa.2012.04.016","article-title":"Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries","volume":"391","author":"Amari","year":"2012","journal-title":"Physica A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ay, N., Jost, J., Le, H.V., and Schwachh\u00f6fer, L. (2017). Information Geometry, Springer.","DOI":"10.1007\/978-3-319-56478-4"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/S0378-4371(02)01018-X","article-title":"Deformed exponentials and logarithms in generalized thermostatistics","volume":"316","author":"Naudts","year":"2002","journal-title":"Physica A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1142\/S0129055X04002151","article-title":"Continuity of a class of entropies and relative entropies","volume":"16","author":"Naudts","year":"2004","journal-title":"Rev. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Naudts, J. (2011). Generalised Thermostatistics, Springer Science & Business Media.","DOI":"10.1007\/978-0-85729-355-8"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1214\/aos\/1176324311","article-title":"An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one","volume":"23","author":"Pistone","year":"1995","journal-title":"Ann. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2032","DOI":"10.1214\/aos\/1176348385","article-title":"Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems","volume":"19","author":"Csiszar","year":"1991","journal-title":"Ann. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1007\/s10959-011-0400-5","article-title":"On \u03d5-Families of probability distributions","volume":"26","author":"Vigelis","year":"2013","journal-title":"J. Theor. Probab."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Ohara, A. (2018). Conformal flattening for deformed information geometries on the probability simplex. Entropy, 20.","DOI":"10.3390\/e20030186"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1088\/0305-4470\/32\/7\/002","article-title":"Maximum entropy approach to stretched exponential probability distributions","volume":"32","author":"Anteneodo","year":"1999","journal-title":"J. Phys. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.physa.2017.12.069","article-title":"Towards an information geometric characterization\/classification of complex systems. I. Use of generalized entropies","volume":"496","author":"Ghikas","year":"2018","journal-title":"Physica A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1140\/epjst\/e2016-60159-x","article-title":"Generalization of the possible algebraic basis of q-triplets","volume":"226","author":"Tsallis","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.physa.2014.05.009","article-title":"Generalized Shannon\u2013Khinchin axioms and uniqueness theorem for pseudo-additive entropies","volume":"411","year":"2014","journal-title":"Physica A"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Jizba, P., and Korbel, J. (2017). On the uniqueness theorem for pseudo-additive entropies. Entropy, 19.","DOI":"10.3390\/e19110605"},{"key":"ref_38","first-page":"223","article-title":"Can the maximum entropy principle be explained as a consistency requirement?","volume":"26","author":"Uffink","year":"1995","journal-title":"Stud. Hist. Philos. Sci. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"6390","DOI":"10.1073\/pnas.1103539108","article-title":"Generalized entropies and the transformation group of superstatistics","volume":"108","author":"Hanel","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/2\/112\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:28:32Z","timestamp":1760185712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/2\/112"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,24]]},"references-count":39,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["e21020112"],"URL":"https:\/\/doi.org\/10.3390\/e21020112","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,1,24]]}}}