RenewlWellcomeTrustCentreforNeuroimaging：更新神经影像学威康课件

AbstractThispresentationreviewsvariationaltreatmentsofdynamicmodelsthatfurnishtime-dependentconditionaldensitiesonthepathortrajectoryofasystem'sstatesandthetime-independentdensitiesofitsparameters.Theseobtainbymaximizingavariationalactionwithrespecttoconditionaldensities.Theactionorpath-integraloffree-energyrepresentsalower-boundonthemodel’slog-evidenceormarginallikelihoodrequiredformodelselectionandaveraging.Thisapproachrestsonformulatingtheoptimizationingeneralizedcoordinatesofmotion.TheresultingschemecanbeusedforonlineBayesianinversionofnonlinearhierarchicaldynamiccausalmodelsandisshowntooutperformexistingapproaches,suchasKalmanandparticlefiltering.Furthermore,itprovidesformultipleinferenceonamodelsstates,parametersandhyperparametersusingexactlythesameprinciples.Free-form(Variationalfiltering)andfixedform(DynamicExpectationMaximization)variantsoftheschemewillbedemonstratedusingsimulated(bird-song)andrealdata(fromhemodynamicsystemsstudiedinneuroimaging).VariationalfilteringandDEMEPSRCSymposiumWorkshopon

ComputationalNeuroscienceMonday8–Thursday11,December2008.OverviewHierarchicaldynamicmodelsGeneralisedcoordinates(dynamicalpriors)Hierarchalforms(structuralpriors)Variationalfilteringandaction(free-form)LaplaceapproximationandDEM(fixed-form)ComparativeevaluationsExamples(HemodynamicsandBirdsongs).Generalisedcoordinates

Likelihood(Dynamical)prior.EnergiesandgeneralisedprecisionsInstantaneousenergyGeneralandGaussianformsPrecisionmatricesingeneralisedcoordinatesandtime.Hierarchicaldynamicmodels.HierarchalformsandempiricalpriorsAsimpleenergyfunctionofpredictionerrorDynamicalpriors(empirical)Structuralpriors(empirical)Priors(full).OverviewHierarchicaldynamicmodelsGeneralisedcoordinates(dynamicalpriors)Hierarchalforms(structuralpriors)Variationalfilteringandaction(free-form)LaplaceapproximationandDEM(fixed-form)ComparativeevaluationsExamples.Free-energy:Expectedenergy:Entropy:Aim:Tooptimisethepath-integral(Action)ofafree-energyboundonmodelevidencew.r.t.arecognitiondensity

qWhenoptimised,therecognitiondensityapproximatesthetrueconditionaldensityandActionbecomesaboundapproximationtotheintegratedlog-evidence;thesecanthenbeusedforinferenceonparametersandmodelspacerespectivelyVariationallearning.WhereandarethepriorenergiesVariationalenergyandactionsRecognitiondensityLemma:ThefreeenergyismaximisedwithrespecttowhenandtheinstantaneousenergyisspecifiedbyagenerativemodelWenowseekrecognitiondensitiesthatmaximiseactionMean-fieldapproximation.EnsemblelearningLemma:isthestationarysolution,inamovingframeofreference,foranensembleofparticles,whoseequationsofmotionandensembledynamicsareThisdescribesastationarydensityunderamovingframeofreference,withvelocity

asseenusingtheco-ordinatetransformProof:SubstitutingtherecognitiondensitygivesVariationalfiltering.-202-505Atoyexample020406080100120-2-1012345.OverviewHierarchicaldynamicmodelsGeneralisedcoordinates(dynamicalpriors)Hierarchalforms(structuralpriors)Variationalfilteringandaction(free-form)LaplaceapproximationandDEM(fixed-form)ComparativeevaluationsExamples(hemodynamicsandbirdsongs).Optimizingfree-energyundertheLaplaceapproximationMean-fieldapproximation:Laplaceapproximation:ConditionalmodesUndertheseapproximations,allweneedtodoisoptimisetheconditionalmodesConditionalprecisionsTheLaplaceapproximationenablesusthespecifythesufficientstatisticsoftherecognitiondensityverysimply.Takingtheexpectationoftheensembledynamics,weget:Here,canberegardedasagradientascentinaframeofreferencethatmovesalongthetrajectoryencodedingeneralisedcoordinates.Thestationarysolution,inthismovingframeofreference,maximisesvariationalaction.bytheFundamentallemma;c.f.,Hamilton'sprincipleofstationaryaction.Approximatingthemode…agradientascentinmovingcoordinates.DynamicexpectationmaximizationAdynamicrecognitionsystemthatminimisespredictionerrorE-SteplearningM-StepuncertaintyD-Stepinferencelocallinearisation(Ozaki1992).OverviewHierarchicaldynamicmodelsGeneralisedcoordinates(dynamicalpriors)Hierarchalforms(structuralpriors)Variationalfilteringandaction(free-form)LaplaceapproximationandDEM(fixed-form)ComparativeevaluationsExamples(hemodynamicsandbirdsongs).PredictionerrorGenerationInversionAlinearconvolutionmodel.Variationalfilteringonstatesandcauses51015202530-1-0.500.511.5hiddenstates51015202530-0.4-0.200.20.40.60.811.2causetime{bins}timecause.Lineardeconvolutionwithvariationalfiltering(SDE)–freeformLineardeconvolutionwithDynamicexpectationmaximisation(ODE)–fixedform.Accuracyandembedding(n)1357911130123456sumsquarederror(causalstates)010203040-1.5-1-0.500.511.52time05101520253035-1.5-1-0.500.511.52timeTheorderofgeneralisedmotionPrecisioningeneralisedcoordinates.010203040-1-0.500.51timehiddenstatesDEM(0)EKFhiddenstatessumofsquarederror(hiddenstates)05101520253035-1.5-1-0.500.51timeDEM(0)DEM(4)EKFtrueDEMandextendedKalmanfilteringEKF0.10.20.30.40.50.60.70.80.9DEM(0)DEM(4)Withconvergencewhen.AnonlinearconvolutionmodellevelThissystemhasaslowsinusoidalinputorcausethatexcitesincreasesinasinglehiddenstate.Theresponseisaquadraticfunctionofthehiddenstates(c.f.,Arulampalametal2002).

.ComparativeperformanceSumofsquarederrorDEMandparticlefiltering.Tripleestimation(DEM)Inferenceonstates

Learningparameters

.OverviewHierarchicaldynamicmodelsGeneralisedcoordinates(dynamicalpriors)Hierarchalforms(structuralpriors)Variationalfilteringandaction(free-form)LaplaceapproximationandDEM(fixed-form)ComparativeevaluationsHemodynamicsandBirdsongs.Stimuli250radiallymovingdotsat4.7degrees/sPre-Scanning5x30strialswith5speedchanges(reducingto1%)Task:detectchangeinradialvelocityScanning(nospeedchanges)4x100scansessions;eachcomprising10scansof4differentconditionsFAFNFAFNS.................A–dots,motionandattention(detectchanges)N–dotsandmotionS–dotsF–fixationBucheletal1999AnfMRIstudyofattentionV5(motionsensitivearea).AhemodynamicmodelstateequationsoutputequationOutput:amixtureofintra-andextravascularsignalconvol