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Entropy2017,19, 7 Acknowledgments: This researchwaspartiallysupportedbyJSPS(JapanSociety for thePromotionofScience), KAKENHI(Grants-in-AidforScientificResearch)GrantNumbers JP26108003andJP15K04842. Conflictsof Interest:Theauthordeclaresnoconflictof interest. References 1. Tsallis,C. Introduction toNonextensiveStatisticalMechanics:ApproachingaComplexWorld; Springer:NewYork, NY,USA,2009. 2. Naudts, J. GeneralisedThermostatistics; Springer: London,UK,2011. 3. Kaniadakis,G. Theoretical foundations andmathematical formalismof thepower-law tailed statistical distributions. Entropy2013,15, 3983–4010. 4. Beck, C.; Schlögl, F. Thermodynamics of Chaotic Systems: An Introduction; CambridgeUniversity Press: Cambridge,UK,1993. 5. Naudts, J. Estimators, escortprobabilities, andφ-exponential families instatisticalphysics. J. Inequal. Pure Appl.Math. 2004,5, 102. 6. Matsuzoe,H.;Henmi,M.Hessianstructuresanddivergence functionsondeformedexponential families. InGeometricTheory of Information, Signals andCommunicationTechnology;Nielsen, F., Ed.; Springer: Basel, Switzerland,2014;pp. 57–80. 7. Sakamoto,M.;Matsuzoe,H.Ageneralizationof independenceandmultivariateStudent’s t-distributions. InGeometricScienceof Information,ProceedingsofSecondInternationalConferenceonGeometricScienceof Information(GSI2015),Palaiseau,France,28–30October2015;Volume9389,pp. 740–749. 8. Matsuzoe,H.;Wada,T. Deformedalgebrasandgeneralizationsof independenceondeformedexponential families. Entropy2015,17, 5729–5751. 9. Wada,T.;Matsuzoe,H.;Scarfone,A.M. Dualistichessianstructuresamongthe thermodynamicpotentials in theκ-Thermostatistics. Entropy2015,17, 7213–7229. 10. Matsuzoe,H. Statisticalmanifoldsandgeometryofestimating functions. InProspects ofDifferentialGeometry andItsRelatedFields,Proceedingsofthe3rdInternationalColloquiumonDifferentialGeomentryandItsRelatedFields, VelikoTarnovo,Bulgaria, 3–7September2012;Adachi,T.,Hashimoto,H.,Hristov,M.J.,Eds.;WorldScientific: Hackensack,NJ,USA,2013;pp. 187–202. 11. Matsuzoe,H.;Henmi,M. Hessian structures ondeformedexponential families. InGeometric Science of Information,ProceedingsofFirst InternationalConferenceonGeometricScienceof Information (GSI2013),Paris, France,28–30August2013; Springer: Berlin/Heidelberg,Germany,2015;Volume8085,pp. 275–282. 12. Eguchi,S.;Komori,O. Pathconnectednessonaspaceofprobabilitydensity functions. InGeometricScience of Information, Proceedings of Second InternationalConference onGeometric Science of Information (GSI 2015), Palaiseau,France, 28–30October2015; Springer: Berlin/Heidelberg,Germany,2015;Volume9389,pp.615–624. 13. Scarfone, A.M.; Matsuzoe, H.; Wada, T. Consistency of the structure of Legendre transform in thermodynamicswith theKolmogorov-Nagumoaverage. Phys. Lett. A2016,380, 3022–3028. 14. Tsallis,C.Whatare thenumbers thatexperimentsprovide? Quim.Nova1994,17, 468–471. 15. Zhang, J. Onmonotoneembedding in informationgeometry. Entropy2015,17, 4485–4499. 16. TanakaM.Meaningofanescortdistributionandτ-transformation. J.Phys. Conf. Ser. 2010,201, 012007. 17. Murata, N.; Takenouchi, T.; Kanamori, T.; Eguchi, S. Information geometry ofU-boost and Bregman divergence. NeuralComput. 2004,16, 1437–1481. 18. Kurose,T. Onthedivergencesof1-conformallyflatstatisticalmanifolds. TôhokuMath. J.1994,46, 427–433. 19. Matsuzoe,H. Statisticalmanifoldsandaffinedifferentialgeometry. Adv. Stud. PureMath. 2010,57, 303–321. 20. Lauritzen,S.L. Statisticalmanifolds. InDifferentialGeometry inStatistical Inferences;Gupta,S.S.,Ed.; IMS Lecture Notes Monograph Series 10; Institute of Mathematical Statistics: Hayward, CA, USA, 1987; pp.96–163. 21. Amari,S.;Nagaoka,H.Methodof InformationGeometry;TranslationsofMathematicalMonographs;American MathematicalSociety: Providence,RI,USA;OxfordUniversityPress:Oxford,UK,2000. 22. Amari,S. InformationGeometryandItsApplications; Springer: Tokyo, Japan,2016. 23. Amari,S.;Ohara,A.;Matsuzoe,H. Geometryofdeformedexponential families: Invariant,dually-flatand conformalgeometry. PhysicaA2012,391, 4308–4319. 323
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Differential Geometrical Theory of Statistics
Title
Differential Geometrical Theory of Statistics
Authors
Frédéric Barbaresco
Frank Nielsen
Editor
MDPI
Location
Basel
Date
2017
Language
English
License
CC BY-NC-ND 4.0
ISBN
978-3-03842-425-3
Size
17.0 x 24.4 cm
Pages
476
Keywords
Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
Categories
Naturwissenschaften Physik
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