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1、JMLR:WorkshopandConferenceProceedings1:1-16GaussianProcessesinPracticeGaussianProcessApproximationsofStochasticDifferentialEquationsC´edricArchambeauc.archambeau@cs.ucl.ac.ukDepartmentofComputerScience,UniversityCollegeLondonGowerStreet,LondonWC1E6BT,UnitedKingdomDanCornford
2、d.cornford@aston.ac.ukNeuralComputingResearchGroup,AstonUniversityAstonTriangle,BirminghamB47ET,UnitedKingdomManfredOpperopperm@cs.tu-berlin.deArtificialIntelligenceGroup,TechnicalUniversityBerlinFranklinstraße.28/29,D-10587Berlin,GermanyJohnShawe-Taylorjst@cs.ucl.ac.ukDepar
3、tmentofComputerScience,UniversityCollegeLondonGowerStreet,LondonWC1E6BT,UnitedKingdomEditor:NeilD.Lawrence,AntonSchwaighoferandJoaquinQui˜noneroCandelaAbstractStochasticdifferentialequationsarisenaturallyinarangeofcontexts,fromfinancialtoenvironmentalmodeling.Currentsolutionm
4、ethodsarelimitedintheirrepresentationoftheposteriorprocessinthepresenceofdata.Inthiswork,wepresentanovelGaussianprocessapproximationtotheposteriormeasureoverpathsforageneralclassofstochasticdifferentialequationsinthepresenceofobservations.Themethodisappliedtotwosimpleproblem
5、s:theOrnstein-Uhlenbeckprocess,ofwhichtheexactsolutionisknownandcanbecomparedto,andthedouble-wellsystem,forwhichstandardapproachessuchastheensembleKalmansmootherfailtoprovideasatisfactoryresult.Experimentsshowthatourvariationalapproximationisviableandthattheresultsareverypr
6、omisingasthevariationalapproximatesolutionoutperformsstandardGaussianprocessregressionfornon-GaussianMarkovprocesses.Keywords:DynamicalSystems,StochasticProcesses,BayesianInference,GaussianProcesses1.IntroductionStochasticdifferentialequationsareusedinawiderangeofapplication
7、sinenvironmentalmodeling,engineeringandbiologicalmodeling.Theytypicallydescribethetimedynamicsoftheevolutionofastatevector,basedonthe(approximate)physicsoftherealsystem,togetherwithadrivingnoiseprocess.Thenoiseprocesscanbethoughtofinseveralways.Itoftenrepresentsprocessesnot
8、includedinthemodel,butpresentintherealsystem.Inourworkwearemotivatedbyproblemsaris
