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1、MaximumLikelihoodEstimatesforGaussianMixturesAreTranscendentalCarlosAmendola1,MathiasDrton2,andBerndSturmfels1;31TechnischeUniversit•at,Berlin,Germany2UniversityofWashington,Seattle,USA3UniversityofCalifornia,Berkeley,USAAbstract.Gaussianmixturemodelsare
2、centraltoclassicalstatistics,widelyusedintheinformationsciences,andhavearichmathematicalstructure.Weexaminetheirmaximumlikelihoodestimatesthroughthelensofalgebraicstatistics.TheMLEisnotanalgebraicfunctionofthedata,sothereisnonotionofMLdegreeforthesemodels
3、.Thecriti-calpointsofthelikelihoodfunctionaretranscendental,andthereisnoboundontheirnumber,evenformixturesoftwounivariateGaussians.Keywords:Algebraicstatistics,expectationmaximization,maximumlikelihood,mixturemodel,normaldistribution,transcendencetheory1I
4、ntroductionTheprimarypurposeofthispaperistodemonstratetheresultstatedinthetitle:Theorem1.ThemaximumlikelihoodestimatorsofGaussianmixturemod-elsaretranscendentalfunctions.Moreprecisely,thereexistrationalsamplesx;x;:::;x2Qnwhosemaximumlikelihoodparametersfo
5、rthemixtureof12Ntwon-dimensionalGaussiansarenotalgebraicnumbersoverQ.Theprincipleofmaximumlikelihood(ML)iscentraltostatisticalinference.MostimplementationsofMLestimationemployiterativehill-climbingmethods,suchasexpectationmaximization(EM).Thesecanrarelyce
6、rtifythatagloballyoptimalsolutionhasbeenreached.Analternativeparadigm,advancedbyalge-braicstatistics[8],isto ndtheMLestimator(MLE)bysolvingthelikelihoodequations.Thisisonlyfeasibleforsmallmodels,butithasthebene tofbeingarXiv:1508.06958v1[math.ST]27Aug2015
7、exactandcerti able.CentraltothisapproachistheMLdegree,whichisde nedasthealgebraicdegreeoftheMLEasafunctionofthedata.Thisrestsonthepremisethatthelikelihoodequationsaregivenbypolynomials.Manymodelsusedinpractice,suchasexponentialfamiliesfordiscreteorGaussia
8、nobservations,canberepresentedbypolynomials.Hence,theyhaveanMLdegreethatservesasanupperboundforthenumberofisolatedlocalmaximaofthelikelihoodfunction,independentlyofthesamplesizeandthedata.TheMLdegreeisanintrinsicinv
