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Entropy2016,18, 407 onhowthegeneralizationofRényidivergence is relatedtoZhang’s (ρ,τ)-divergence,andalsohow thepresentproposal is relatedto themodelpresented in [33]. Acknowledgments: Theauthors are indebted to the anonymous reviewers for their valuable comments and corrections, which led to a great improvement of this paper. Charles C. Cavalcante also thanks the CNPq (Proc.309055/2014-8) forpartial funding. AuthorContributions:Allauthorscontributedequally to thedesignof theresearch. Theresearchwascarried outbyallauthors. RuiF.VigelisandCharlesC.Cavalcantegavethecentral ideaof thepaperandmanagedthe organizationof it. RuiF.Vigeliswrote thepaper.All theauthorsreadandapprovedthefinalmanuscript. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. References 1. Rao,C.R. Informationandtheaccuracyattainable in theestimationofstatisticalparameters. Bull. Calcutta Math. Soc. 1945,37, 81–91. 2. Amari, S.-I. Differential geometry of curved exponential families—Curvatures and information loss. Ann.Stat. 1982,10, 357–385. 3. Amari, S.-I. Differential-Geometrical Methods in Statistics; Springer: Berlin/Heidelberg, Germany, 1985; Volume28. 4. Amari, S.-I.; Nagaoka, H. Methods of Information Geometry (Translations of Mathematical Monographs); AmericanMathematicalSociety: Providence,RI,USA,2000;Volume191. 5. Amari, S.-I. InformationGeometry and ItsApplications; AppliedMathematical Sciences Series; Springer: Berlin/Heidelberg,Germany,2016;Volume194. 6. Amari,S.-I.;Ohara,A.;Matsuzoe,H. Geometryofdeformedexponential families: Invariant,dually-flatand conformalgeometries. PhysicaA2012,391, 4308–4319. 7. Matsuzoe, H. Hessian structures on deformed exponential families and their conformal structures. Differ.Geom.Appl. 2014,35 (Suppl.), 323–333. 8. Naudts, J. Estimators, escortprobabilities, andφ-exponential families instatisticalphysics. J. Inequal. Pure Appl.Math. 2004,5, 102. 9. Pistone,G. κ-exponentialmodels fromthegeometricalviewpoint. Eur. Phys. J.B2009,70, 29–37. 10. Amari, S.-I.;Ohara,A. Geometryof q-exponential familyofprobabilitydistributions. Entropy2011, 13, 1170–1185. 11. Vigelis,R.F.;Cavalcante,C.C. TheΔ2-Conditionandϕ-FamiliesofProbabilityDistributions. InGeometric Scienceof Information; Springer: Berlin/Heidelberg,Germany,2013;Volume8085,pp. 729–736. 12. Vigelis,R.F.;Cavalcante,C.C. Onϕ-familiesofprobabilitydistributions. J.Theor. Probab. 2013,26, 870–884. 13. Cena,A.;Pistone,G. Exponential statisticalmanifold. Ann. Inst. Stat.Math. 2007,59, 27–56. 14. Grasselli,M.R. Dualconnections innonparametricclassical informationgeometry. Ann. Inst. Stat.Math. 2010,62, 873–896. 15. Pistone, G.; Sempi, C. An infinite-dimensional geometric structure on the space of all the probability measuresequivalent toagivenone. Ann. Stat. 1995,23, 1543–1561. 16. Santacroce,M.; Siri, P.; Trivellato, B. Newresults onmixture andexponentialmodels byOrlicz spaces. Bernoulli2016,22, 1431–1447. 17. Vigelis,R.F.;Cavalcante,C.C. InformationGeometry:AnIntroduction toNewModels forSignalProcessing. InSignals and Images;CRCPress: BocaRaton,FL,USA,2015;pp. 455–491. 18. Vigelis, R.F.; de Souza, D.C.; Cavalcante, C.C. New Metric and Connections in Statistical Manifolds. InGeometricScienceof Information; Springer: Berlin/Heidelberg,Germany,2015;Volume9389,pp. 222–229. 19. Zhang, J. Divergence function,duality,andconvexanalysis. NeuralComput. 2004,16, 159–195. 20. Zhang, J. ReferentialDualityandRepresentationalDualityonStatisticalManifolds. In Proceedingsof the 2ndInternationalSymposiumonInformationGeometryandItsApplications,Pescara, Italy,12–16December 2005;pp. 58–67. 21. Zhang, J. Nonparametric informationgeometry: Fromdivergence functiontoreferential-representational bidualityonstatisticalmanifolds. Entropy2013,15, 5384–5418. 22. Zhang, J. DivergenceFunctionsandGeometricStructuresTheyInduceonaManifold. InGeometricTheoryof Information; Springer: Berlin/Heidelberg,Germany,2014;pp. 1–30. 285