(2006 ppt)Advances in Gaussian Processes

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1、AdvancesinGaussianProcessesTutorialatNIPS2006inVancouverCarlEdwardRasmussenMaxPlanckInstituteforBiologicalCybernetics,TübingenDecember4th,2006Rasmussen(MPIforBiologicalCybernetics)AdvancesinGaussianProcessesDecember4th,20061/55ThePredictionProblem420400?380360concentr

2、ation,ppm2340CO3201960198020002020yearRasmussen(MPIforBiologicalCybernetics)AdvancesinGaussianProcessesDecember4th,20062/55ThePredictionProblem420400380360concentration,ppm2340CO3201960198020002020yearRasmussen(MPIforBiologicalCybernetics)AdvancesinGaussianProcessesDe

3、cember4th,20063/55ThePredictionProblem420400380360concentration,ppm2340CO3201960198020002020yearRasmussen(MPIforBiologicalCybernetics)AdvancesinGaussianProcessesDecember4th,20064/55ThePredictionProblem420400380360concentration,ppm2340CO3201960198020002020yearRasmussen

4、(MPIforBiologicalCybernetics)AdvancesinGaussianProcessesDecember4th,20065/55ThePredictionProblemUbiquitousquestions:•Modelfitting•howdoIfittheparameters?•whataboutoverfitting?•ModelSelection•howtoIfindoutwhichmodeltouse?•howsurecanIbe?•Interpretation•whatistheaccuracyofth

5、epredictions?•canItrustthepredictions,evenif•...Iamnotsureabouttheparameters?•...Iamnotsureofthemodelstructure?Gaussianprocessessolvesomeoftheabove,andprovideapracticalframeworktoaddresstheremainingissues.Rasmussen(MPIforBiologicalCybernetics)AdvancesinGaussianProcess

6、esDecember4th,20066/55OutlinePartI:foundationsPartII:advancedtopics•WhatisaGaussianprocess•Example•fromdistributiontoprocess•priorsoverfunctions•distributionoverfunctions•hierarchicalpriorsusing•themarginalizationpropertyhyperparameters•Inference•learningthecovariance

7、function•Bayesianinference•posterioroverfunctions•Approximatemethodsfor•predictivedistributionclassification•marginallikelihood•GaussianProcesslatentvariable•Occam’sRazormodels•automaticcomplexitypenalty•SparsemethodsRasmussen(MPIforBiologicalCybernetics)AdvancesinGaus

8、sianProcessesDecember4th,20067/55TheGaussianDistributionTheGaussiandistributionisgivenby
p(x

9、µ,Σ)=N(µ,Σ)=(2π)−D/2

10、Σ

11、−1/2exp