MethodsofInformationGeometry

MethodsofInformationGeometry
Informationgeometryprovidesthemathematicalscienceswithanewframeworkofanalysis.Ithasemergedfromtheinvestigationofthenaturaldifferentialgeometricstructureonmanifoldsofprobabilitydistributions,whichconsistsofaRiemannianmetricdefinedbytheFisherinformationandaone-parameterfamilyofaffineconnectionscalledthe$\alpha$-connections.Thedualitybetweenthe$\alpha$-connectionandthe$(-\alpha)$-connectiontogetherwiththemetricplayanessentialroleinthisgeometry.Thiskindofduality,havingemergedfrommanifoldsofprobabilitydistributions,isubiquitous,appearinginavarietyofproblemswhichmighthavenoexplicitrelationtoprobabilitytheory.Throughtheduality,itispossibletoanalyzevariousfundamentalproblemsinaunifiedperspective.Thefirsthalfofthisbookisdevotedtoacomprehensiveintroductiontothemathematicalfoundationofinformationgeometry,includingpreliminariesfromdifferentialgeometry,thegeometryofmanifoldsorprobabilitydistributions,andthegeneraltheoryofdualaffineconnections.Thesecondhalfofthetextprovidesanoverviewofwideareasofapplications,suchasstatistics,linearsystems,informationtheory,quantummechanics,convexanalysis,neuralnetworks,andaffinedifferentialgeometry.Thebookwillserveasasuitabletextforatopicscourseforadvancedundergraduatesandgraduatestudents. 本软件ID:1623634