gaussian_processes_in_machine_learning

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1、GaussianProcessesinMachineLearningGerhardNeumann,SeminarF,WS05/06OutlineofthetalkGaussianProcesses(GP)[ma05,rs03]BayesianInferenceGPforregressionOptimizingthehyperparametersApplicationsGPLatentVariableModels[la04]GPDynamicalModels[wa05]GP:IntroductionG

2、aussianProcesses:Definition:AGPisacollectionofrandomvariables,anyfinitenumberofwhichhavejointGaussianDistributionDistributionoverfunctions:GaussianDistribution:overvectorsNonlinearRegression:XN…DataPointstN…TargetVectorInferNonlinearparameterizedfuncti

3、on,y(x;w),predictvaluestN+1fornewdatapointsxN+1E.g.FixedBasisFunctionsBayesianInferenceoftheparametersPosteriorpropabilityoftheparameters:Probabilitythattheobserveddatapointshavebeengeneratedbyy(x;w)OftenseparableGaussiandistributionisusedEachdatapoint

4、tidifferingfromy(xi;w)byadditivenoisepriorsontheweightsPredictionismadebymarginalizingovertheparametersIntegralishardtocalculateSampleparameterswfromthedistributionwithMarkovchainMonteCarlotechniquesOrApproximatewithaGaussianDistributionBayesianInferen

5、ce:SimpleExampleGP:isaGaussiandistributionExample:HFixedBasisfunctions,NinputpointsPrioronw:Calculatepriorfory(x):Priorforthetargetvaluesgeneratedfromy(x;w)+noise:CovarianceMatrix:CovarianceFunctionPredictingDataInfertN+1giventN:Simple,becausecondition

6、aldistributionisalsoaGaussianUseincrementalformofWecanrewritethisequationUsepartitionedinverseequationstogetfromPredictivemean:UsuallyusedfortheinterpolationUncertaintyintheresult:PredictingDataBayesianInference:SimpleExampleHowdoesthecovariancematrixl

7、ooklike?UsuallyN>>H:Qhasnotfullrank,butChas(duetotheadditionofI)SimpleExample:10RBFfunctions,uniformlydistributedovertheinputspaceBayesianInference:SimpleExampleAssumeuniformlyspacedbasisfunctions,SolutionoftheintegralLimitsofintegrationtoMoregeneralfo

8、rmGaussianProcessesOnlyCNneedstobeinverted(O(N³))PredictiondependentirelyonCandtheknowntargetstNGaussianProcesses:CovariancefunctionsMustgenerateanon-negativedefinitecovariancematrixforanysetofpointsHyperparametersofCSomeExamples:RBF:Li