On the geometry of generalized Gaussian distributi

arXiv:0706.0606v1[math.PR]5Jun2007OnthegeometryofgeneralizedGaussiandistributions∗AttilaAndai†RIKEN,BSI,AmariResearchUnit2–1,Hirosawa,Wako,Saitama351-0198,Japan.May11,2007AbstractInthispaperweconsiderthespaceofthoseprobabilitydistributionswhichmaximizetheq-R´enyientropy.Thesedistributionshavethesameparameterspaceforeveryq,andintheq=1casethesearethenormaldistributions.SomemethodstoendowthisparameterspacewithRiemannianmetricispresented:thesecondderivativeoftheq-R´enyientropy,Tsallis-entropyandtherelativeentropygiverisetoaRiemannianmetric,theFisher-informationmatrixisanaturalRiemannianmetric,andtherearesomegeometricallymotivatedmetricswhichwerestudiedbySiegel,CalvoandOller,Lovri´c,Min-OoandRuh.Thesemetricsaredifferentthereforeourdifferentialgeometricalcalculationsbasedonaunifiedmetric,whichcoversalltheabovementionedmetricsamongothers.Wealsocomputethegeometricalpropertiesofthismetric,theequationofthegeodesiclinewithsomespecialsolutions,theRiemannandRiccicurvaturetensorsandscalarcurvature.Usingthecorrespondencebetweenthevolumeofthegeodesicballandthescalarcurvatureweshowhowtheparameterqmodulatesthestatisticaldistinguishabilityofclosepoints.Weshowthatsomefrequentlyusedmetricinquantuminformationgeometrycanbeeasilyrecoveredfromclassicalmetrics.1IntroductionIntheoreticalstatisticsandinapplicationsthedistancefunctionsbetweenprobabilitydistributionsplayanimportantrole.Theconstructionofaproperdistancefunctionhasbeenconsideredbyseveralauthors.Buteventhesamestatisticalmodelwithdifferentmathematicalframeworkscanleadtodifferentdistancefunctions.Tonarrowthefamilyofpotentialdistancefunctionsweconsiderthosewhicharenaturalfromdifferentialgeometricalpointofview.HistoricallythepioneeringworkofMahalanobis[23]wasgeneralizedbyRao[30],whofirstsuggestedtheideaofconsideringtheFisherinformation[14]asaRiemannianmetriconthespaceofprobabilitydistributions.Cencov[8]wasthefirsttostudymonotonemetricsonstatisticalmanifolds.Heprovedthat,uptoanormalization,thereexistsauniquemonotonemetric,theFisherinformation.Amari[3]andAmariandNagaoka[4]providemodernaccountofthegeneraldifferentialgeometrythatarisesfromtheFisherinformationmetric.TheFishermetricwasstudiedfurtherbyAkin[1],James[16],Burbea[6],Mitchell[22],AtkinsonandMitchell[5],Skovgaard[34],Oller[25],OllerandCuadrasa[27],OllerandCorcuera[26]amongotherresearchers.Thecombinationofdifferentialgeometricalandstatisticalstudieshelpedtofindthestatisticalinterpretationofgeometricalquantities.Forexamplethegeodesicdistancebetweenprobabilitydistributions,whichisusuallyknownasRaodistanceisanaturaldistancefunctionbetweenprobabilitydistributions;thestatisticalmeaningoftheso-called∗keywords:Gaussiandistribution,differentialgeometry;MSC:94A17,53B21†[e-mail]1e-curvaturewasfirstclarifiedbyEfron[12];thenormalizedvolumemeasureofthemanifoldiscalledJeffreys’prior[17]withinthefieldofBayesianstatistics.Inthispaperweconsiderthespaceofthoseprobabilitydistributionswhichmaximizetheq-R´enyientropy.Thesedistributionshavethesameparameterspaceforeveryq,andintheq=1casethesearethenormaldistributions.ThefirstresultsaboutthegeometricalpropertiesofthesespacesareduetoAmari[3,2].HeconsideredtheFisherinformationmetriconthesemanifoldsandcomputedsomegeometricalinvariants.SomemethodstoendowtheparameterspacewithRiemannianmetricispresented:thesecondderivativeoftheq-R´enyientropy[31],Tsallis-entropy[35]andtherelativeentropygiverisetoaRiemannianmetric,theFisher-informationmatrixisanaturalRiemannianmetric,andtherearesomegeometricallymotivatedmetricswhichwerestudiedbySiegel[33],CalvoandOller[7]andLovri´c,Min-OoandRuh[32].Thesemetricsaredifferentthereforeourdifferentialgeometricalcalculationsbasedonaunifiedmetric,whichcoversalltheabovementionedmetricsamongothers.Wealsocomputethegeometricalpropertiesofthismetric,theequationofthegeodesiclinewithsomespecialsolutions,theRiemannandRiccicurvaturetensorsandscalarcurvature.Usingthecorrespondencebetweenthevolumeofthegeodesicballandthescalarcurvatureweshowhowtheparameterqmodulatesthestatisticaldistinguishabilityofclosepoints.Weshowthatsomefrequentlyusedmetricinquantuminformationgeometrycanbeeasilyrecoveredfromclassicalmetrics.2q-R´enyientropymaximizingdistributionsThenormaldistributionscanbeintroducedasaresultofthemaximumentropyprinciple.Considerthefamilyofdensityfunctionswhicharecontinuousandsupportedonthereallinewithgivenexpectationvalueμ∈Randvarianceσ2∈R.IntroducingtheLagrangemultipliersa,b,cwehavethefollowingfunctionalonthefamilyofprobabilitydistributionsS(p)=−Zp(x)logp(x)dx−aZp(x)dx−1−bZp(x)xdx−μ−cZp(x)(x−μ)2dx−σ2.ThevariationofthefunctionalisδS=Z−logp(x)−1−a−bx−c(x−μ)2
δp(x)dx.Thefunctionalhasextremalpointatpifitsvariationiszero.Onecanshowthattheentropyfunctionalhaslocalmaximumatthepointp(x)=exp−a−bx−c(x−μ)2
forappropriateparametersa,b,c∈R.ThefamilyofonedimensionalnormaldistributionsS1canbeparameterizedbytheexpectationvalueu∈Randtheparameterd∈R+asf(d,u,x)=√d√2πe−12d(x−u)2.ThismeansthatS1canbeidentifiedwitha2dimensionalspaceΞ1=R+×R.ThestatisticalpropertiesofthedistributionsleadustodefineRiemannianmetricon