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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
