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A gentle tutorial on the EM algorithm and its application to parameter estimation for gauss

We describe the maximum-likelihood parameter estimation problem and how the ExpectationMaximization form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the

AGentleTutorialoftheEMAlgorithm

anditsApplicationtoParameter

EstimationforGaussianMixtureand

HiddenMarkovModelsJeffA.Bilmes(bilmes@cs.berkeley.edu)

InternationalComputerScienceInstitute

BerkeleyCA,94704

and

ComputerScienceDivision

DepartmentofElectricalEngineeringandComputerScience

U.C.Berkeley

TR-97-021

April1998

Abstract

Wedescribethemaximum-likelihoodparameterestimationproblemandhowtheExpectation-Maximization(EM)algorithmcanbeusedforitssolution.We rstdescribetheabstractformoftheEMalgorithmasitisoftengivenintheliterature.WethendeveloptheEMpa-rameterestimationprocedurefortwoapplications:1) ndingtheparametersofamixtureofGaussiandensities,and2) ndingtheparametersofahiddenMarkovmodel(HMM)(i.e.,theBaum-Welchalgorithm)forbothdiscreteandGaussianmixtureobservationmodels.Wederivetheupdateequationsinfairlyexplicitdetailbutwedonotproveanyconver-genceproperties.Wetrytoemphasizeintuitionratherthanmathematicalrigor.

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