Stephan, KE, Schlagenhauf, F., Huys, QJM, Raman, S., Aponte, EA, Brodersen, KH, ... Heinz, A.

21 Pages·2017·1.94 MB·English

Stephan, K. E., Schlagenhauf, F., Huys, Q. J. M., Raman, S., Aponte,

E. A., Brodersen, K. H., Rigoux, L., Moran, R. J., Daunizeau, J.,
Dolan, R. J., Friston, K. J., & Heinz, A. (2016). Computational
neuroimaging strategies for single patient predictions. NeuroImage,
145(B), 180-199. https://doi.org/10.1016/j.neuroimage.2016.06.038
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NeuroImage145(2017)180–199
ContentslistsavailableatScienceDirect
NeuroImage
journal homepage: www.elsevier.com/locate/ynimg
Computational neuroimaging strategies for single patient predictions
K.E.Stephana,b,c,F.Schlagenhaufd,e,Q.J.M.Huysa,f,S.Ramana,E.A.Apontea,K.H.Brodersena,L.Rigouxa,c,
R.J.Moranb,g,J.Daunizeaua,h,R.J.Dolanb,i,K.J.Fristonb,A.Heinzd,j
aTranslationalNeuromodelingUnit(TNU),InstituteforBiomedicalEngineering,UniversityofZurich&ETHZurich,8032Zurich,Switzerland
bWellcomeTrustCentreforNeuroimaging,UniversityCollegeLondon,London,WC1N3BG,UK
cMaxPlanckInstituteforMetabolismResearch,50931Cologne,Germany
dDepartmentofPsychiatryandPsychotherapy,CampusCharitéMitte,Charité-UniversitätsmedizinBerlin,10115Berlin,Germany
eMaxPlanckInstituteforHumanCognitiveandBrainSciences,04130Leipzig,Germany
fDepartmentofPsychiatry,PsychosomaticsandPsychotherapy,HospitalofPsychiatry,UniversityofZurich,Switzerland
gVirginaInstituteofTechnology,USA
hICMParis,France
iMaxPlanckUCLCentreforComputationalPsychiatryandAgeingResearch,London,UK
jHumboldtUniversitätzuBerlin,BerlinSchoolofMindandBrain,10115Berlin,Germany
a r t i c l e i n f o a b s t r a c t
Articlehistory: Neuroimagingincreasinglyexploitsmachinelearningtechniquesinanattempttoachieveclinicallyrelevant
Received27November2015 single-subjectpredictions.Analternativetomachinelearning,whichtriestoestablishpredictivelinksbetween
Revised21May2016 featuresoftheobserveddataandclinicalvariables,isthedeploymentofcomputationalmodelsforinferring
Accepted20June2016
onthe(patho)physiologicalandcognitivemechanismsthatgeneratebehaviouralandneuroimagingresponses.
Availableonline22June2016
Thispaperdiscussestherationalebehindacomputationalapproachtoneuroimaging-basedsingle-subjectinfer-
ence,focusingonitspotentialforcharacterisingdiseasemechanismsinindividualsubjectsandmappingthese
Keywords:
characterisationstoclinicalpredictions.Followinganoverviewoftwomainapproaches–Bayesianmodelselec-
Generativemodel
fMRI tionandgenerativeembedding–whichcanlinkcomputationalmodelstoindividualpredictions,wereviewhow
EEG thesemethodsaccommodateheterogeneityinpsychiatricandneurologicalspectrumdisorders,helpavoiderro-
Bayesian neousinterpretationsofneuroimagingdata,andestablishalinkbetweenamechanistic,model-basedapproach
Modelselection andthestatisticalperspectivesaffordedbymachinelearning.
Modelcomparison ©2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense
Modelevidence (http://creativecommons.org/licenses/by/4.0/).
Generativeembedding
Classification
Clustering
Computationalpsychiatry
Translationalneuromodeling
Introduction However,indirectapplicationtoclinicalquestions,neuroimaging-
directedmachinelearninghasmainlybeenusedtodiscriminatepatient
Despiteitspotentialtoprovideanon-invasiveassayofwhole-brain groupsfromeachotherorfromhealthycontrols.Thishasbeenvariably
function,neuroimaginghasexperiencedsurprisingdifficultiesindeliv- successful,asindicatedbythediverseoutcomesfromfMRI-basedclas-
eringdiagnosticapplicationsforclinicalpractice,particularlyinpsychi- sification competitions. In these competitions, the accuracies of
atry.Whilethereareseveralreasonsforthisdisappointingtrackrecord, neuroimaging-baseddiagnoseshaverangedfrompoor(e.g.,attention
whichissharedbyotherapproacheslikegenetics(fordiscussionsand deficit hyperactivity disorder; Brown et al., 2012) to excellent
reviews,seeCaseyetal.,2013;Kapuretal.,2012;KrystalandState, (e.g.,schizophrenia;Silvaetal.,2014).Moreimportantly,however,
2014;Stephanetal.,2015),theperceivedfailurehastriggeredimpor- theattempttoreplaceoraugmenttraditionalclinicaldiagnosticsbyap-
tantdiscussionsaboutthemostpromisingavenuesforclinicalneuroim- plyingmachinelearningtechniquestoneuroimagingdataisastrategy
aging.Oneparticularhopeisthattheinfluxofmethodsfrommachine oflimitedlong-termclinicalutility.Thisisbecauseamultitudeofphys-
learningwillrealisethetranslationalpotentialofneuroimaging.Forex- iological,geneticandclinicalstudiesoverthepastdecadeshavemadeit
ample,infMRI,multivariateclassificationandregressionschemeshave clearthatmentaldiseasesasdefinedbycontemporaryclassification
recentlyyieldedimpressivesuccessesinclinicallyrelevantdomains schemes–suchastheDiagnosticandStatisticalManualofMentalDis-
such as pharmacological or pain research (e.g., Duff et al., 2015; orders(DSM)ortheInternationalClassificationofDiseases(ICD)–are
Wageretal.,2013). highlyheterogeneous.Thatis,disorderslikeschizophrenia,depression,
http://dx.doi.org/10.1016/j.neuroimage.2016.06.038
1053-8119/©2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).
K.E.Stephanetal./NeuroImage145(2017)180–199 181
autismetc.grouppatientswithsimilarclustersofsymptomsandsigns numberofkeyfeatures,whichwediscussindetailbelow.Inshort,com-
thatarecausedbydiversepathophysiologicalmechanisms(Cuthbert putationalapproaches(i)allowonetoinvokemodelcomparisonproce-
andInsel,2013; Kapuretal.,2012;Krystal andState,2014; Owen, duresforclarifyingwhetheranyneurophysiologicaldifferencesamong
2014;Stephanetal.,2016).Thispathophysiologicaldiversityexplains patientssignalrealdifferencesinpathophysiology,orsimplyreflectdif-
whypsychiatricdiagnosesbasedonDSM/ICDhavelittlepredictiveva- ferentcognitivestrategies;(ii)providetheory-drivendimensionality
lidity;thatis,withfewexceptions(suchasdifferentiatingmono-and reduction;and(iii)cansupportpowerfulsingle-subjectpredictions
bipolaraffectivedisorders)theydonotinformtheclinicianaboutindi- basedoninferredmechanisms,asopposedtopatternsofdatafeatures.
vidualclinicaltrajectoriesortreatmentresponses.Asaconsequence, Thispaperprovidesanoverviewofcomputationalneuroimaging
evenifhighlyaccurateDSM/ICDdiagnosescouldbederivedfromma- strategiesforsingle-subjectpredictions,independentlyof–orincon-
chinelearningclassifiersappliedtoneuroimagingdata,thiswouldsim- junctionwith–machinelearningtechniques.Weattempttoillustrate
ply recapitulate a diagnostic scheme that does not directly inform centralconceptsofgenerativemodelling,andtheclinicalutilitythey
clinicalmanagement–andwoulddosousingaconsiderablymoreex- mayafford.Tothisend,wefocusonthegeneralformofmodelclasses;
pensiveandlesswidelyavailabletechnologycomparedtoclassicalpsy- bycontrast,wedonotdiscussmathematicalpropertiesofanysingle
chiatricinterviews. modelindetail,butreferthereadertotherelevantliterature.One
Thisisonereasonwhytheapplicationofmachinelearningtoneuro- areawhichrequiresaslightlymoredetailedmathematicaldiscussion
imagingdatahaschangeddirectioninrecentyearsandisnowbeingin- istheframeworkofBayesianmodelcomparison.Evenhere,however,
creasingly applied to problems more directly related to clinical we restrict our treatment to interpreting the general form of key
management,suchaspredictingindividualdiseasecourseortreatment equations.
efficacy.Thishasshownsomepromisingrecentresults,indicatingthat Thispaperhasthefollowingstructure.First,forreaderswithout
itmaybecomepossibletopredictindividualtrajectoriesofpatients muchbackgroundincomputationalneuroimaging,weprovideabrief
withschizophrenia(Anticevicetal.,2015)ormooddisorders(Lythe overviewofexistingapproaches,clarifysomenomenclature,andrevisit
etal.,2015;Schmaaletal.,2015)fromneuroimagingdata,orforecast someofitsprevioussuccesses.Second,wediscusstheimportanceof
individual treatment responses to psychotherapy (Mansson et al., modelcomparisonfordealingwithheterogeneityacrossindividuals
2015),antidepressants(DeBattistaetal.,2011;McGrathetal.,2013; andintroducetheprinciplesofBayesianmodelselection(BMS)and
Milleretal.,2013)andantipsychotics(Hadleyetal.,2014;Nejadetal., Bayesianmodelaveraging(BMA).Third,weoutlinehowclinicalpredic-
2013). tionscanbederivedfromcomputationalmodels,(i)illustratingtheuse
Theseareremarkablesuccessesandraisehopesthatneuroimaging ofBMSwhentheoriesofdiseasemechanismsexistand(ii)introduc-
may finally contribute to clinical decision-making in the not-too- inggenerativeembeddingasalinkbetweencomputationalmodel-
distantfuture.However,thestraightforwardapplicationofmachine lingandmachinelearning,whendisease(process)theoriesarenot
learningtoneuroimagingdatafacesanumberofnon-trivialtechnical available.Moreover,weoutlinehowBMSandgenerativeembedding
andconceptualchallengesthatmayimpedetheirlong-termclinical canbedeployedinanunsupervisedorsupervisedfashioninorderto
utility(fordiscussionsandreviews,seeBrodersenetal.,2011;Klöppel addressproblemsrelatedtonosology(i.e.,detectingsubgroupsin
etal.,2012;Orruetal.,2012;Wolfersetal.,2015).Onecentralchallenge heterogeneous disorders), differential diagnosis, and outcome
isthatneuroimagingdataarenoisyandveryhigh-dimensional,pre- prediction.
sentingwithamyriadofdatafeaturesthatcouldinformprediction.Di-
mensionality reduction and optimal feature selection thus become Computationalneuroimaging–what,whyandhow?
criticalproblems.Oneinterestingdevelopmentinthisregardconcerns
recent advances in feature extraction methods based on restricted Theterm“computational”originallyderivesfromthetheoryofcom-
Boltzmannmachines(Hjelmetal.,2014)anddeepneuralnetworks putation,asubfieldofmathematicsthatexamineswhichparticular
(Plisetal.,2014),whichoffernovelrepresentationsofdiseasestates functionsarecomputable.Afunctioniscomputableifitrepresentsa
withpotentialdiagnosticopportunities. mapping(fromaninputtoanoutputset)thatcanbeimplementedby
Second,hemodynamicandelectrophysiologicalmeasurementsrep- analgorithm;i.e.,awell-definedsequenceofoperations.Inneurosci-
resentdistalandpotentiallycomplicatedtransformsofunderlyingneu- ence,theterm“computational”hasbeenusedquiteflexibly,ranging
ronalmechanisms.Thismeansthatconventionalmachinelearning fromanemphasisoninformationprocessing(irrespectiveofitsbio-
methods,whichoperatedirectlyonobservedfeaturesofneuroimaging, physicalimplementation),toverybroadusage,encompassinganyalgo-
donotfurnishmechanisticinsightsintopathophysiology.Thiscanbeil- rithmic investigation of neuronal systems (cf. “computational
lustratedwiththreeexamples.First,inmultivariateclassificationstud- neuroscience”),incontrasttoanalyticalmathematicaltreatments.
iesoffMRI,eventhoughthespatialdistributionofinformativevoxels Inneuroimaging,threemaincomputationalapproachesarepresent-
canbedetermined,thisdoesnotdiscloseaconcretebiologicalprocess. lybeingpursued(forreview,seeStephanetal.,2015).Theseinclude
Similarly,unsupervisedlearningapproachesthatexploitmultimodal biophysicalnetworkmodels(BNMs),generativemodels,and“model-
imaging measures can obtain compelling subgroup delineations basedfMRI”.BNMsarelarge-scalenetworkmodelswhosenodesrepre-
(Ingalhalikaretal.,2014)butremaindescriptiveanddonotoffera sentmeanfield(orneuralmass)modelsofneuronalpopulationactivi-
mechanisticlinkbetweenthestructuralandfunctionalcomponentsof ty.Augmentedwithahemodynamicorelectrophysiologicalforward
any identified predictor. Finally, while functional connectivity model,neuronalpopulationactivityistranslatedintoapredictedfMRI
(i.e.,statisticaldependenciesbetweenregionaltimeseries)hasenabled or EEG signal (for reviews, see Deco et al., 2013a; Deco and
successfulmachinelearningapplications(Arbabshiranietal.,2013; Kringelbach,2014;WangandKrystal,2014).Byconnectingtheindivid-
Craddocketal.,2009;Duetal.,2015;Richiardietal.,2011;Rosaetal., ualnodesinaccordancewithanatomicalconnectivitydata–obtained
2015),itscharacterisationofneuronalprocessesisrestrictedtostatisti- fromhumandiffusion-weightedimagingorMacaquetracttracingstud-
calcorrelationsthatareagnosticaboutthephysiologicalcausesofnet- ies–thedynamicsoflarge-scalenetworksandensuingwhole-brain
work dynamics. In general, machine learning applied to “raw” neuroimagingsignalscanbesimulated.Whiletheneuronalstateequa-
neuroimagingdatadoesnoteasilyidentifymechanismsfromwhich tionsofBNMscanberichinbiophysicaldetail,theircomplexityrenders
noveltherapeuticapproachescouldbederived. parameterestimationverydifficult,andcurrentmodelsallowonlythe
Analternativetomachinelearningistheuseoftheory-drivencom- estimationofasingleglobalscalingparameterofconnectivity(Deco
putationalmodelstoinferpathophysiologicalmechanismsinindividual etal.,2013b).Forthisreason,thispaperfocusesontheothertwoclasses
patients(Fristonetal.,2014;Huysetal.,2016;MaiaandFrank,2011; ofmodels,generativemodelsandmodel-basedfMRI.Theserestonless
Montagueetal.,2012;StephanandMathys,2014).Thisstrategyhasa complexandfine-grainedformulations,butallowforestimatingmodel
182 K.E.Stephanetal./NeuroImage145(2017)180–199
Fig.1.Summaryofagenerativemodelforneuroimagingdata.Thisfigurecontainsgraphicsthatarereproduced,withpermission,fromChenetal.(2009),Garridoetal.(2008),and
Stephanetal.(2003).
parametersfrommeasureddata.Inthefollowing,wesummarisethe lowercaseitalics;sets,functionalsandquantitiesfromprobabilitytheo-
principlesofthesemodelsandclarifysomeofthetechnicaltermsand ry(suchasinformation-theoreticsurpriseorfreeenergy)byuppercase
conceptsinvolved.Thefollowingconventionswillbeusedforequa- italics, vectors by lowercase bold, and matrices by uppercase bold
tions:functions,distributionsandscalarvariablesarerepresentedby letters.
Fig.2.OverviewofDCMforfMRI.Reproduced,withpermission,fromStephanetal.(2015).
K.E.Stephanetal./NeuroImage145(2017)180–199 183
Generativemodelsofneuroimagingdata inference schemes. These comprise two main approaches: Markov
Instatistics,a“generativemodel”isdefinedbythejointprobability chainMonteCarlo(MCMC)samplingandvariationalBayes(VB);for
overallrandomvariables(e.g.,observeddataandmodelparameters) in-depthdiscussionsseeMacKay(2003)andBishop(2006).MCMCis
thatdefineasystemofinterest.Moreintuitively,onecanthinkofagen- computationallyexpensiveandcanrequireverylongruntimesbutis
erativemodelasdescribinghowobserveddataweregeneratedand guaranteedtoconvergetothecorrectsolution(inthelimitofinfinite
henceviewingitasa“recipe”forgeneratingsimulateddata.Agenera- time).Ontheotherhand,VBiscomputationallyveryefficientbutissus-
tivemodelisspecifiedbydefiningtwocomponents:alikelihoodfunc- ceptibletolocalextremaandcanbeaffectedbyviolationsofdistribu-
tion and a prior density. The likelihood function rests on a tionalassumptions(Daunizeauetal.,2011).
probabilisticmappingfromhiddenquantities(parametersθ)ofthesys- Inthispaper,wefocusonthefirstandmostwidelyusedgenerative
temofinteresttoobservablequantities(measurements)y: modellingframeworkforneuroimagingdata,dynamiccausalmodelling
(DCM).ThisapproachwasintroducedadecadeagoforfMRI(Friston
y¼fðθÞþε ð1Þ etal.,2003).DCMforfMRIusesdifferentialequationstodescribethedy-
namicsofneuronalpopulationstatesx(t)thatinteractviasynapticcon-
Thissimplysaysthatthedata(feature)vectoryoriginatesfrom nectionsandaresubjecttoexperimentallycontrolledperturbations
sometransformationf,whichencodesaputativesignal-generatingpro- u(t).Theseperturbationscaneitherinduceneuronalpopulationactivity
cess,plusstochasticnoiseε.WedeliberatelywriteEq.1inthisform,asit directly;e.g.,intermsofsensorystimulation(“drivinginput”)ordy-
willprovideausefulreferenceforthedistinctionbetweenfeatureselec- namicallymodulatethestrengthsofsynapticconnections.Theformof
tionmethodsthatdoordonotexploitknowledgeaboutthehidden theseneuronalstateequationsisgivenbyalow-order(Taylor)approx-
causesofmeasurements(seesectiononGenerativeEmbeddingbelow). imationtoanynonlinearsystem(Fristonetal.,2003;Stephanetal.,
Thefunctionfencodeshowsystemparametersdetermineitsoutput 2008),wherethestrengthsofsynapticconnectionsandweightsofdriv-
(signal);thiscanrangefromextremelysimpleconcepts(e.g.aconstant ingandmodulatoryinputsrepresenttheparametersofinterestwewish
termdescribingmeansignal)tocomplexfunctions;e.g.,abiophysically toinferbymodelinversion(seeFig.2).1
motivateddynamicalsystem,asinthecaseofdynamiccausalmodelling Theactivityofeachneuronalpopulationiscoupledtoaregional
(DCM,seebelow).Eq.1cannowbeusedtospecifythelikelihoodfunc- bloodoxygenleveldependent(BOLD)signalbyacascadeofdifferential
tionasquantifyingtheprobabilityp(y|θ)ofobservingaparticularmea- equationsdescribinghemodynamicprocesses,suchaschangesinblood
surement y, given a particular parameterisation of the system. For flowandbloodvolume(Fristonetal.,2000;Stephanetal.,2007).2Tech-
example,assumingidenticallyandindependentlydistributedGaussian nically,thismeansthatDCMrepresentsahierarchicalgenerativemodel,
noiseε,thelikelihoodcanbewrittenas: wherethelikelihoodfunctionispartitionedintodeterministicdynamic
(cid:1) (cid:3) stateequationsfofhiddenneuronalandhemodynamicprocesses(with
pðyjθÞ¼N y;fðθÞ;σ2I ð2Þ neuronalandhemodynamicparametersθ ,θ )andastaticobservation
n h
functiongthatimplementsdiscretesamplingandaccountsformea-
whereIdenotestheidentitymatrixandσ2noisevariance. surementnoiseε(forthedetailedequations,seeFig.2andFriston
Thepriordensityencodestherangeofvaluestheparametersareex-
etal.,2003;Stephanetal.,2007):
pectedtotakeapriori,i.e.,beforeanydataareobserved.Again,under
Gaussianassumptionswecanexpressthisas:
dx
Neuronalstates: n¼ f ðx ;u;θ Þ
pðθÞ¼Nðθ;μθ;ΣθÞ ð3Þ ddtx n n n ð5Þ
Hemodynamicstates: h¼f ðx ;x ;θ Þ
whereμθ,Σθdenotepriormeanandpriorcovariance,respectively. Measurements: ydt¼gðxhÞþn εh h
h
Togeneratedata,onecouldsimplysamplefromthepriordensity
andplugtheensuingparametervaluesintothelikelihoodfunction.
Notably,intheseequations,noiseonlyentersattheobservation
Thisapproachtosimulatingdataisverygeneral,andtraditionalnon-
probabilisticsimulationsincomputationalneuroscience(withfixedpa- levelwhereasneuronaldynamicsunfoldsinadeterministicfashion,
givenexternalperturbationsandsystemparameters(e.g.,synapticcon-
rameters)canberegarded asaspecialcase of agenerative model,
nectionstrength).Thatis,inEq.5,thedynamicvariablesofinterest
wherethepriordensityreducestoaDiracdeltafunction(pointmass)
(neuronalandhemodynamicstatesx ,x )aredeterministicfunctions
overthechosenparameters. n h
ofdesignedandknowninputsuandoftime-invariantneuronal(θ )
Usingtheprobabilisticmappingfromhiddenparametersofthesys- n
temtoobservedsignalsinthe“forward”directionisveryusefulforsim- andhemodynamic(θh)parameters.(Foreaseofreading,wehaveomit-
tedexplicitreferencestotime.)Thismeansthatweonlyneedtoinfer
ulatingobservableresponses,underdifferentparametervaluesand
theparameters–thestatetrajectoriesfollowautomaticallyfromanyin-
exploringsystembehaviour.Inneuroimaging,however,wewishto
ferredparametervalues.Thiswouldbedifferentifstochasticdifferential
proceedintheoppositedirection;i.e.,estimatetheparametervalues
equationswerechosen;inthiscase,thestatesarenotfullydetermined
fromobserveddata.Thisreversemappingisequivalentlyreferredto
as“modelinversion”,solvingthe“inverseproblem”,orsimply“infer- bythechoiceofparametersandwouldneedtobeinferred,inaddition
ence”.Formally,thiscorrespondstocomputingtheposteriorprobability totheparameters.ThisisknownasstochasticDCM(Daunizeauetal.,
p(θ|y)=N(θ;μθ|y,Σθ|y)oftheparameters,giventhedata(Fig.1).This 2009;Daunizeauetal.,2012;Lietal.,2011).
Byspecifyingplausiblepriordensitiesovertheneuronalandhemo-
followsdirectlyfromBayestheorem:
dynamicparameters(seeFristonetal.,2003fordetails),thegenerative
pðyjθ;mÞpðθjmÞ modeliscompleted.Invertingthisgenerativemodelallowsonetoinfer
pðθjy;mÞ¼ ð4Þ
pðyjmÞ theneuronalparametersofinterest(e.g.,couplingstrengthsandtheir
modulationbyexperimentalconditions)fromempiricallymeasured
Here,wehavemadethedependencyonachosenmodelstructurem fMRI data. Notably, the separate representation of neuronal and
explicitbyconditioningalltermsonm.Thepracticaldifficultyisthatde-
rivingtheterminthedenominatorrequirescomputinganintegral(see 1 SimulationsthatprovideanintuitionofneuronaldynamicsaccountedforbyDCMand
Eq.9)whichisusuallynotanalyticallytractable,exceptforsomevery illustratehowdifferentparametersimpactontheresultingsignalscanbefoundinseveral
previouspapers;forexample,seeFig.1inPennyetal.(2004b)andFig.2inStephanetal.
simplecases.Unfortunately,evennumericalintegrationisrarelyfeasi-
(2008).
ble,sincecomputationtimeincreasesexponentiallywiththenumber 2 Forsimulationsillustratingthenatureofthishemodynamicmodel,seeFigs.3,8,9in
ofmodelparameters.Inpractice,onehastoresorttoapproximate Fristonetal.(2000)andFig.5inStephanetal.(2007).
184 K.E.Stephanetal./NeuroImage145(2017)180–199
hemodynamicmechanismsiscrucialforgenerativemodelsoffMRI; Themodel-basedfMRIapproachwaspioneeredbyO'Dohertyand
sincevariabilityinneurovascularcouplingacrossregionsandsubjects colleagues(O'Dohertyetal.,2003)whousedatemporaldifference
canotherwiseconfoundinferenceonconnectivity(Davidetal.,2008). (TD)learningmodeltoshowthatphasicactivationoftheventralstria-
FollowingitsintroductionforfMRI,DCMhasbeenextendedtoother tum,amajorrecipientofdopaminergicprojectionsfromthemidbrain,
neuroimagingmodalities,includingevent-relatedpotentials(ERPs; correlatedwiththemagnitudeoftrial-wiserewardPEsduringanin-
Davidetal.,2006),inducedresponses(Chenetal.,2008),andspectral strumentalconditioningtask.Thiswasmotivatedbytheseminalwork
responses(Moranetal.,2009),asmeasuredbyelectroencephalography ofSchultz,Dayan,andMontaguewhofoundthatrewardPEscorrelated
(EEG)andmagnetoencephalography(MEG).DCMhasfoundwide- withphasicactivityofdopaminergicmidbrainneurons(Schultzetal.,
spreaduseforanalysisofeffectiveconnectivitybetweenneuronalpop- 1997)andthatchangesinthisphasicactivityduringlearningcouldbe
ulationsandhasfurnishedinsightsintocircuit-levelmechanismsthat predicted under the formalism of a TD learning model (see also
eluded previous schemes.This includes,for example, physiological (Montagueetal.,2004)forreview).
characterisationsofpredictivecodingincorticalhierarchiesduringper- Model-basedfMRIhassubsequentlybeenappliedtonumerousdo-
ceptualinferenceandlearning,bothinhealthysubjects(e.g.,denOuden mainsofcognition,accommodatingadiversityofmodellingapproaches
etal.,2009;Garridoetal.,2008;Summerfieldetal.,2006)andinpa- andcomputationalthemes,suchasreinforcementlearning(RL)models
tients with schizophrenia (Dima et al., 2010; Dima et al., 2009; ofbehaviourandBayesianmodelsofcognition(e.g.,(D'Ardenneetal.,
Ranlundetal.,2015)oralteredlevelsofconsciousnessduetobrain 2013;Dawetal.,2006;Iglesiasetal.,2013;Klein-Flüggeetal.,2011;
damage.Beyondlong-rangeconnectivity,DCMhasalsoprovenuseful Schwartenbecketal.,2015;Seymouretal.,2004;Vosseletal.,2015).
forinferringdetailed,low-levelphysiological(synaptic)mechanisms Arecentapplicationofmodel-basedfMRIhasbeentheinvestigation
withinlocalneuronalcircuitsofthehumanbrain,exploitingtherich of interactions between learning and decision-making processes
temporalinformationcontainedbyEEG/MEGdata.Examplesinclude whichdoordonotderivefromanexplicitmodeloftheenvironment
thedetectionofconductancechangesinAMPAandNMDAreceptors ortaskstructure.Thisdistinctioniscommonlyreferredtoas“model-
underdopaminergicmodulation(Moranetal.,2011),changesinpost- based”vs.“model-free”computations(e.g.,Dawetal.,2011;Deserno
synaptic gain of supragranular pyramidal cells in auditory cortex etal.,2015;Gläscheretal.,2010;Huysetal.,2012);wherethelatter
undercholinergicstimulation(Moranetal.,2013),orthecharacterisa- terminducesaterminologicaltwistinthecontextofmodel-basedfMRI.
tionofchangesinneuronalphysiologyinindividualswithselectivemu- Model-basedfMRIinhumanshasproducedresultsofhighrelevance
tationsofparticularionchannels(Gilbertetal.,2016). for pathophysiological theories, corroborating, for example,
hypothesisedlinksbetweentrial-by-trialactivityinneuromodulatory
Model-basedfMRI nuclei and the trajectories of specific computational quantities
Generativemodelsalsoplayacentralroleinthesecondcomputa- suggestedbytheoreticalaccountsand/oranimalexperiments.Promi-
tionalapproachconsideredinthispaper,“model-basedfMRI”.Howev- nent examples include the encoding of reward PEs and precision
er,incontrasttothepurelyphysiologicalDCMsdescribedabove,this (inverseuncertainty)byphasicandtonicchangesinactivitylevelsof
approachaskswhethera(particularcomponentofa)computational thedopaminergicmidbrain(e.g.,D'Ardenneetal.,2013;Klein-Flügge
processisreflectedinBOLDsignals(GläscherandO'Doherty,2010; etal.,2011;Schwartenbecketal.,2015),orthereflectionofexpected
O'Dohertyetal.,2003).Inotherwords,ittriestoexplainvoxel-wise andunexpecteduncertainty(YuandDayan,2005)byactivityinthe
BOLDsignalsasalinearmixtureofcomputationalprocesses,which cholinergicbasalforebrain(Iglesiasetal.,2013)andnoradrenergic
areassumedtobedirectlyencodedbytheunderlyingneuronalactivity. locuscoeruleus(Payzan-LeNestouretal.,2013).Model-basedfMRI
Thesameapproachcanbeapplied,ofcourse,toM/EEGresponses;for has also been applied to patients, for example, in depression
example,inorderexplaintrial-by-trialamplitudesorwaveformsof (Dombrovskietal.,2013;Gradinetal.,2011)andaddiction(Harle
eventrelatedpotentials(Liederetal.,2013;Ostwaldetal.,2012).How- etal.,2015;Tanabeetal.,2013).Perhapsmostnotably,model-based
ever,givenitsdominanceinthepresentcomputationalneuroimaging fMRI studies of patients with schizophrenia (Gradin et al., 2011;
literature,weherefocusentirelyonmodel-basedfMRI. Murrayetal.,2008;Romaniuketal.,2010)havecontributedempirical
Model-basedfMRIrestsonatwo-stepprocedure(seeFig.2A).First, evidence for the long-standing hypothesis that disturbances in PE
agenerativemodelofbehaviouralresponses,withcomputationalstates signalling by dopaminergic neurons in the midbrain might assign
x (e.g.,trial-wisepredictionerrors(PEs)oruncertainty)andparame- “aberrant salience” to environmental events (Heinz, 2002; Kapur,
c
tersθ,areestimatedusingthemeasuredbehavioury ofanindividual: 2003).
c b
yb¼gðxc;θcÞþεb ð6Þ Hybridandunifiedmodels
Therelationbetweenthebehaviouralandneuroimagingdomainsof
Byinvertingthismodel,thecomputationalstatesxccanbeinferred model-based fMRI is summarised schematically in Fig. 3A. This
(dottedlinemarkedwith1inFig.2A).Thesubsequentconvolutionwith highlights the fact (expressed by equations 6 and 7 above) that
astandardhemodynamicresponsefunction(HRF)thenprovidesex- model-basedfMRIessentiallyrepresentsacorrelationalapproachbe-
planatoryvariablesorregressorsforastandardmass-univariateGLM tweentwotypesofmeasurements,eachofwhichhasitsowngenera-
analysisofvoxel-wisefMRIdata: tiveprocess.Inthelongrun,unifiedmodelsmaybedevelopedthat
y ¼ðx ⊗HRFÞβþε ð7Þ explainbothneuroimagingsignalsandbehaviourofagivenindividual
fMRI c fMRI fromthesameunderlyingstateequation(i.e.,neuronalprocess).Forex-
ample,thiscouldbeastateequationdescribingthebiophysicalimple-
TheparametersofthisGLM,β,cannowbeestimatedinasecondin-
mentationofrelevantcomputationsinacircuitofinterest;thiswould
ferencestep(dottedlinemarkedwith2inFig.2A),eitherunderflat
requirebothamappingfrom(hidden)neuronalstatestobehavioural
priors(i.e.,maximumlikelihoodestimation)orusingempiricalBayes-
observationsthatconsidersthebiophysicalimplementationofrelevant
ianprocedures(FristonandPenny,2003).Overall,thistwo-stepproce-
computations(e.g.,predictivecoding),andamappingfromcircuitstate
dureenablesonetosearch,acrossthewholebrain,fortheneuronal
toneuroimagingdatadescribinghowneuronalactivitytranslatesinto
correlatesofcomputationalvariablesofinterestwhichhadbeenin-
measurablesignals(Fig.3B).Thiswouldallowonetoinfercircuitpa-
ferredfromthesimultaneouslymeasuredbehaviour.3
rametersofinterest,simultaneouslyfrombothmeasuredbehaviour
andneuroimagingdataandprovideamechanistic(andquantitative)
3 Foraveryinstructiveoverviewwithbothsimulationsandempiricalresults,pleasesee characterisation ofneuronalprocessingthatwasgroundedbothin
Figures1and2inGlascherandO'Doherty(2010). termsofphysiologyandcomputationalfunction.Moreover,unified
K.E.Stephanetal./NeuroImage145(2017)180–199 185
Fig.3.A.Summaryofthetwo-stepprocedurein“model-based”fMRI.Greyplatesdenoterandomvariables;lightblueellipsesrepresentobservedvariables(measurements);darkerblue
diamondsrepresentstates(whichfollowdeterministically,giventheparametersandinputs).Solidarrowsrepresentdependenciesamongvariables;dashedarrowsrepresentinference.B.
Summaryofaunifiedmodelinwhichbothbehaviouralandneuroimagingdataarepredictedfromthesameunderlyingstateequation.Inthismodel,theunknownparameterscanbe
inferredinonestep,basedonbothbehaviouralandneuroimagingdata.Seemaintextfordetailsanddefinitionofvariables.
modelsofthissortcanhelpidentifyingwhichpartsofacircuitarepar- (seetheirFigs.3–5forsimulationsthatdemonstratetheprinciplesof
ticularlycriticalformaladaptiveactions.Thismightbeparticularlyuse- thismodel).Whilethisrelativelycoarsestateequationonlyallowsfor
ful when addressing the problem of multiple potential causes or relativelysimplisticrelationsbetweencircuitdynamicsandbehaviour,
strategiesunderlyinganobservedbehaviour;cf.(Schlagenhaufetal., theconceptualadvanceofthismodelisconsiderablesinceithelpsiden-
2014). tifyingwhichpartsofthenetwork(nodesorconnections)arecrucialfor
Asignificantsteptowardssuchaunifiedmodelhasbeenmadere- funnellinginputs(stimuliortaskinstructions)intobehaviouraloutputs.
cently(RigouxandDaunizeau,2015).Theyproposedamappingfrom Priorworktowardsintegratinggenerativemodelsofneurophysiolo-
asinglestateequationofcircuitdynamics(basedontheformalismof gyandcomputation(informationprocessing)havemainlyexamined
DCMforfMRI)tosimultaneouslyacquiredbehaviourandfMRIsignals thenotionofPEsas“teachingsignals”thatregulatetheamountof
Fig.4.Illustrationthatmodelselectioncanprovideaformalbasisfordifferentialdiagnosis.Here,therelativeplausibilityofasetofcompetingmodels,representingalternativemechanisms
howtheobserveddatacouldhavebeengenerated,isevaluatedintermsoftheposteriormodelprobability.Inthetypicalcaseofaflatprioronmodelspace,thelatterisidenticaltothe
modelevidence(seemaintextfordetails).
186 K.E.Stephanetal./NeuroImage145(2017)180–199
(Adamsetal.,2013;Stephanetal.,2009a):(i)alteredprefrontalinputs
thattargetmidbrainneuronsviaNMDAreceptors;(ii)enhancedinputs
fromcholinergicbrainstemnuclei(PPT/LDT),or(iii)alteredautoregula-
tionofdopaminergicmidbrainneurons(byparacrinereleaseofdopa-
mineandactivationofdopaminergicautoreceptors).Disambiguating
betweenthesepossibilities,bycomparingmodelsthatembodythe
abovemechanisms(givenmeasurementsfromthemidbrainandthe
areasitcommunicateswith),wouldhavetremendousrelevanceforde-
lineatingschizophreniaintopathophysiologicalsubgroups–andfor
guidingindividualtreatmentdecisions.
Inwhatfollows,weunpackthestatisticalbasisofdifferentialdiag-
nosisbymodelselection.WehopetofamiliarisethereaderwithBayes-
ian techniques for comparing and selecting models that are used
frequently in the currentliterature andprovide apowerful way to
deal with individual variability in physiology and/or computation.
ThesetechniquesareequivalentlyreferredtoasBayesianmodelcom-
parison(BMC)orBayesianmodelselection(BMS);whiletheformeris
themoregeneralterm,thelatterdescribesthecommonsituationof
selectingasingle(mostplausible)modelfromasetofalternatives.4
Fig.5.Anillustrationofthetrade-offbetweenmodelfitandmodelcomplexity,andan
exampleofoverfitting.Here,modelsofincreasingcomplexityarefittedtodatathat Modelevidence
weregeneratedfromanexponentialfunction,plusaddedobservationnoise.Itcanbe Generally,thefirststepofanymodel-driven(hypothesis-led)inves-
seenthatahighlycomplexmodelfitsthedataperfectlybut,becauseitistryingto tigationistodecidewhichclassofexplanationaccountsfortheobserva-
explainthenoiseaswell,makespredictions(suchasthepronouncedbumpsinthe tions.Thisiswhatallscientistsimplicitlydowhentestinghypotheses–
middleofthedataseries)whichwillnotgeneraliseacrossfutureinstantiationsofthe
althoughthisstepmightnotalwaysrepresentanexplicitchoice.Tech-
datafromthesameunderlyingprocess(“overfitting”).Reproduced,withpermission,
fromPittandMyung(2002). nically,hypothesistestingormodelcomparisoncorrespondstodefining
ahypothesissetormodelspaceMofcompetingexplanationsthatare
deemedplausibleapriori.Thisisequivalenttospecifyingapriorover
synapticplasticityneededtoupdateneuralcircuitsduringlearning. models;where,typically,allmodelswithinMareconsideredequally
SpecificallydenOudenetal.(2009,2010)demonstratedthatshort- likelyandallotherpossiblemodelshavezeropriorprobability:
termplasticityduringsensorylearningcouldbemeasuredbyinferring (cid:4)
howeffectiveconnectionstrengthsweremodulatedbytrial-by-trial pðmÞ¼ 1=jMjifm∈M ð8Þ
predictionerrorsobtainedfromRLandhierarchicalBayesianmodels, 0ifm∉M
respectively.Asimilardemonstrationwasprovidedinthecontextof
learningunderaversiveoutcomes(Royetal.,2014).Mostrecently, (Here,|M|referstothecardinalityofthehypothesisset.)Thechal-
Vosseletal.(2015)showedhowattentionalshiftswereaccompanied lengethenistofindthemodelmwithinMthatprovidesthebestexpla-
bychangesincortical-subcorticalnetworkconnectivitythatevolvedac- nationoftheobserveddata.Importantly,selectingamodelthat“best
cordingtotrial-wiseestimatesofcertainty(precision)oftargetpredic- explains”thedataisnotsimplyastatementaboutmodelfit.Indeed,it
tions,wherethelatterwereinferredfromsaccadiceyemovementdata istrivialtofind,foranydataset,modelswithexcellentorevenperfect
usingahierarchicalGaussianfilter(Mathysetal.,2011). fit;forexample,foranyobservationconsistingoftdatapoints,apolyno-
mialfunctionofordert−1willfitthedataperfectly.Theseoverlyaccu-
Bayesianmodelselection rate models simply explain noise or random fluctuations that are
specifictotheparticularmeasurementanddonotgeneralisetoother
AsoutlinedintheIntroduction,manyconventionallydefinedneuro- (e.g.,future)measurementsofthesameprocess.Thistendencyofan
logicalandprobablyallpsychiatricdiseasesarehighlyheterogeneous: overlyflexiblemodeltorecognisespuriouspatternsinnoiseisreferred
patientswithsimilarsymptomsandbehaviourmaydifferconsiderably toas“overfitting”(seeFig.5ofthispaperandFig.1.4ofBishop(2006)
intermsof(patho)physiologicalmechanismsand/orcognitiveprocess- forexamples).Ontheotherhand,thesimplestmodelpossible,which
es.Acentralgoalforneuroimagingapproacheswiththeambitionof wouldconsistofaconstanttermonlyandexplainsnosignalvariance
clinicalutilityisthustoidentify,inanygivenindividualpatient,the (i.e.,R2=0),canindeedbethebestexplanationofatimeseries–
mostlikelymechanismthatunderliesaparticularobservation(brain whenthetimeseriescontainsnosignalandonlynoise.Insummary,
activitypattern).Thisissimplythechallengeofdifferentialdiagnosis, measuresoffitaloneareinadequatetojudgemodelgoodness(Pitt
whichisubiquitousthroughoutmedicine.Differentialdiagnosismaps andMyung,2002).Instead,thechallengeistoselectmodelsthatgener-
directlyontohypothesistestingwhich,inparametricstatistics,corre- alisebest;thesearethemodelsthatprovideanoptimalbalancebe-
spondstotheformalcomparisonofdifferentmodelsofhowobserved tweenfit(accuracy)andcomplexity.
datacouldhavebeengenerated(Fig.4).Inotherwords,anelegantap- ThisbalanceisimplicitintheBayesianmodelevidenceusedduring
proachtoestablishingdifferentialdiagnosesinpsychiatrybasedonneu- Bayesianmodelselection(BMS).Themodelevidenceistheprobability
roimaging would be to formalise competing pathophysiological ofobservingthedataygiventhemodelm.Thisprobabilityisalsore-
theoriesintermsofalternativegenerativemodels.Therelativeplausi- ferredtoasthemarginalorintegratedlikelihoodandcorrespondsto
bilityofthesemodels(hypotheses)wouldthenbeevaluatedbyformal thedenominatorfromBayestheorem(seeEq.4).Itcanbecomputed
modelcomparisonprocedures,givenempiricalmeasurementsofneu-
roimagingand/orbehaviour. 4 WhilethispaperfocusesontheconceptualandmathematicalfoundationsofBMS,
Asaconcreteexample,manypathophysiologicalconceptsofschizo- previouspapershaveprovidedtoyexamples(simulations)andstep-by-stepBMSanaly-
phreniaconvergeonthenotionofdysregulationofdopaminergicmid- sesofsinglesubjectdatawhichmaybeusefulfortheinterestedreader.Forexample,for
simulations,pleaseseeFigures4–6andTables2–5inPennyetal.(2004a)andFigure2
brain neurons in patients with schizophrenia (Heinz, 2002; Kapur,
inStephanetal.(2009b);fordetailedsinglesubjectBMSanalyses,pleaseseeFigures7–
2003;Kingetal.,1984;Winton-Brownetal.,2014).Thisdysregulation 9andTables6–13inPennyetal.(2004a)andFigures3–4andTable1inStephanetal.
couldbecausedbyatleastthreedifferentmechanisms(fordetails,see (2005).
K.E.Stephanetal./NeuroImage145(2017)180–199 187
byintegratingout(ormarginalising)theparametersfromthejoint ofdatapoints:
probability:
Z AIC¼ logpðyjθ;mÞ−k
logn ð12Þ
pðyjmÞ¼ pðyjθ;mÞpðθjmÞdθ ð9Þ BIC¼ logpðyjθ;mÞ−k
2
ThisiswhytheBayesianmodelevidenceisalsoreferredtoasthe TheadditionalscalingfactorinBICmeansthat,oncethatmorethan
marginallikelihood.Asasimplifyingintuition,theevidencecanbeun- nN8datapointsareavailable,BICentailsastrongercomplexitypenalty
derstoodasprovidingananswertothequestion:“IfIrandomlysampled thanAIC.ThesimplicityoftheircomplexityapproximationsmakesAIC/
frommypriorandpluggedtheresultingvalueintothelikelihoodfunc- BICeasytocompute,buthastwosignificantdisadvantages:AIC/BICig-
tion,howclosewouldthepredicteddatabe–onaverage–tomyob- noreinterdependenciesamongmodelparameters(whichareubiqui-
serveddata?” tous in biological systems; Gutenkunst et al., 2007) nor can they
Inpractice,modelcomparisondoesnotutilisethemodelevidence capturedifferencesinpriorvarianceacrossparameters.
directlybuttypicallyemploysitslogarithm(logevidence).Giventhe Theseissuesareresolvedbyathirdapproximationtothelogevi-
monotonicnatureofthelogarithmicfunction,rankingmodelsbased dence,thenegativefreeenergyF.Itsnamederivesfromcloseconnec-
oneithermodelevidenceorlogevidenceyieldsidenticalresults.How- tions between free energy concepts in statistical physics and
ever,thelogevidenceisnumericallyeasiertodealwith(thelogarithm variationalapproachestoprobabilitytheory(seeFristonetal.,2007;
ofasmallnumberbetweenzeroandoneisalargenegativenumber) NealandHinton,1998).Variationalfreeenergyrepresentsalower
andresultsinmoreintuitiveequations,someofwhichweencounter boundapproximationtothelogevidence,wherethetightnessofthe
below.Italsooffersanadditionalniceintuitionderivedfrominforma- bounddependsonhowwellthetrue(butunknown)posteriorcanbe
tiontheory.Specifically,giventhat(Shannon)surpriseSisdefinedas matchedbyanapproximateposteriorq(ofknownform):
negativelogprobability,foranagentoperatingunderagivenmodel
m,thelogevidencecorrespondstothenegativesurpriseaboutobserv- logpðyjmÞ¼FþKL½qðθÞ∥pðθjy;mÞ(cid:2) ð13Þ
ingthedatay:
Here,KLreferstoKullback-Leiblerdivergenceorrelativeentropy,an
logpðyjmÞ¼−SðyjmÞ ð10Þ informationtheoreticmeasureofthedissimilaritybetweentwoproba-
bilitydensities.TheKLdivergenceiszerowhenthedensitiesareidenti-
Putsimply,logevidence–andhencemodelgoodness–increases calandbecomesincreasinglypositivethemorethetwodensitiesdiffer
whenwearelesssurprisedaboutthedataencountered. (KullbackandLeibler,1951).Importantly,sincewedonotknowthe
Whilethestatisticalprocedureofmodelcomparisontypicallyrests trueposterior,theKLtermcannotbeevaluateddirectly.However,by
onthelogevidence,theresultofcomparingtwomodelsisunderstood maximisingFoneimplicitlyminimisestheKLterm,thustightening
moreintuitivelywhenreportedasaBayesfactor(BF);thisissimply thelowerboundapproximationtothelogevidence(seeFig.6).Thisis
theratiooftwomodelevidences.Aswithp-valuesinfrequentiststatis- achievedbyoptimisingtheapproximateposteriorq(e.g.,whenqis
tics,conventionsexistaboutwhichthresholdsaremeaningfulforBayes Gaussian,findingthemeanandvarianceofqthatmaximisesFaccord-
factors(KassandRaftery,1995).Forexample,aBayesfactorlargerthan ingtoEq.13above).Inotherwords,bymaximisingFwecanbothobtain
20(equivalenttoalogevidencedifferencelargerthan3)wouldbecon- anapproximationtothelogevidenceandtheposteriordensitiesofthe
sideredas“strong”evidenceinfavourofonemodelrelativetoanother. parameters.
Analternativeoption,whenreportingtheresultsofmodelcompar- The negative free energy (and hence model evidence) can be
isonsinanintuitivelyaccessibleform,istocompute,foreachmodelm, decomposedintothefollowingbalancebetweenmodelfitandmodel
i
itsposteriorprobability.Inthetypicalcasewheretheprioronmodelsis complexity(fordetails,seePennyetal.,2004a;Stephanetal.,2007):
uninformativeorflat(cf.Eq.8),thissimplifiestonormalisingtheevi-
denceforeachmodelbythesumofallmodelevidences: F¼hlogpðyjθ;mÞiq−KL½qðθÞ∥pðθjmÞ(cid:2) ð14Þ
pðmjyÞ¼ pðyjmiÞpðmiÞ ¼ pðyjmiÞ ð11Þ Inthisexpression,thefirsttermrepresentsaccuracy:theexpected
i XjMj (cid:1) (cid:3) (cid:1) (cid:3) XjMj (cid:1) (cid:3) loglikelihood,underachosenapproximateposteriorq.Thesecond
p yjm p m p yjm
j j j termrepresentscomplexityandisgivenbyanotherKLdivergence;
j¼1 j¼1 thistimebetweentheapproximateposteriorandtheprior.Whenthe
formoftheapproximateposterioristhesameasthetrueposterior,
Thismakesiteasytoseethatacrossallmodels,theposteriormodel thecomplexityisexactlythedifferencebetweentheposteriorand
probabilitysumstounity. prior and inference becomes exact. Put simply, this means that a
modelhashighcomplexityifitissufficientlyflexibletoallowforasub-
Approximationstothelogevidence stantialbeliefupdate,i.e.,apronounceddivergenceoftheposterior
Onemajorbarriertocomputingthemodelevidenceisthattheinte- fromthepriorbelief.Anotherheuristicisthatthecomplexityreflects
gralinEq.9canrarelybeevaluatedanalytically;furthermore,numerical boththeeffectivenumberofparametersthatneedtobedisplaced
integrationistypicallyprohibitivelyexpensive.Therefore,oneusually fromtheirpriorvaluestoprovideanaccurateexplanationfordataand
resortstoapproximationsofthelogevidence,suchastheAkaikeinfor- thedegreeoftheirdisplacement.
mationcriterion(AIC;Akaike,1974),Bayesianinformationcriterion Thisheuristiccanbeturnedintoamoreformalperspectivebyexam-
(BIC; Schwarz,1978),or negative free energy (Fristonet al.,2007; iningtheanalyticalexpressionofthecomplexitytermunderanas-
NealandHinton,1998;Pennyetal.,2004a).Theseapproximationsde- sumed distributional form for the approximate posterior (see
composemodelgoodnessintoabalanceoftwoterms–accuracyand discussionsinStephanetal.(2009b)andPenny(2012)).Forexample,
complexity.Allofthemagreeinthedefinitionofaccuracyasloglikeli- underGaussianassumptions:
hood.Bycontrast,theydifferconsiderablyintheirapproximationof
complexity.
AICandBIChaveaseeminglystraightforwardapproximationof KL½qðθÞ∥pðθjmÞ(cid:2)¼
cfroemepplaerxaimtye.tIenrsA;ICincBoImC,ptlheixsitisyasdimdiptiloyncaolrlyresscpaolenddsbytoththeelongumnubmerbeorf 12logðdetðCθÞÞ−12log(cid:1)det(cid:1)Cθjy(cid:3)(cid:3)þ(cid:5)μθjy−μθ(cid:6)TC−θ1(cid:5)μθjy−μθ(cid:6) ð15Þ
188 K.E.Stephanetal./NeuroImage145(2017)180–199
Fig.6.Agraphicalillustrationofthenegativefreeenergyapproximationtothelogmodelevidence,anditsevolutionduringmodelinversioninthecontextofvariationalBayes.Here,by
adjustingtheparametersoftheapproximateposteriorsuchthatthenegativefreeenergyFismaximised,oneimplicitlyminimisestheKLdivergencebetweentheapproximateandtrue
posteriorandtightenstheboundonthelogevidence.Seemaintextfordetails.
Here,CθandCθ|ydenotepriorandposteriorcovariancematrices,and superior to BIC for model comparison of directed acyclic graphical
detreferstothedeterminant,amatrixpropertythatcanbeinterpreted models(BealandGhahramani,2003).
asameasureof“volume”(thespacespannedbytheeigenvectorsofthe Analternativetotheaboveapproximationsaresampling-basedap-
matrix).Thisvolumeincreaseswiththenumberofdimensions(the proaches,typicallybasedonMCMC.One(highlysimplified)wayto
rankofthecovariancematrix),andwiththelengthandorthogonality thinkofMCMC–inthisparticularcontext–isofreconstructinganinte-
ofthebasisvectors.Withthisinmind,thefirstterminEq.15means gralbyanapproximation(essentiallylikeahistogram)thatis“experi-
thatcomplexityincreaseswiththenumberoffreeparameters,the enced” by a random walk. Here, each step only depends on the
moreflexibletheseparametersare(thehighertheirpriorvariance), previousone(Markovchain)andtendstomoveinadirectionthatis
andthemoreorthogonaltheyare.Thesecondtermmeansthatcom- likelytoprovideameaningfulcontributiontotheintegral.Depending
plexitydecreaseswithincreasingorthogonalityoftheposteriorparam- onhowmuchcomputationtimeoneiswillingtoinvest,differentop-
eterestimates(adesirablepropertyofaninterpretablemodel)andwith tionsexist.Acomputationallylessexpensiveapproachistouseasingle
increasingposteriorvariances(highlypreciseposteriorestimatesresult chainforobtainingsamplesfromaparticulardistributionandusing
inbrittlemodelpredictionswhichareunlikelytogeneralise).Finally, thesesamplestoevaluateEq.9,inordertoobtainanapproximation
thethirdtermcapturesourheuristicaboveandexpressesthatcomplex- tothemodelevidence.Forexample,usingthechaintosamplefrom
itygrowsthemoretheposteriormeandivergesfromthepriormean. thepriorleads to thepriorarithmetic mean (PAM) approximation
Whilealloftheaboveapproximationshaveprovenusefulinprac- (whichtendstounderestimatethemodelevidence),whereassampling
tice,theycomewithdifferentprosandcons.5AICandBICareeasyto fromtheposteriordistributionleadstotheposteriorharmonicmean
computesincetheloglikelihoodisalwaysavailableandestimating (PHM) approximation (which tends to overestimate the model
complexityboilsdowntosimplycountingthenumberoffreeparame- evidence).
ters.Onthedownside,AICandBICareagnostictoseveralimportantas- Amorerobustalternativeismulti-chainsampling.Thekeyideahere
pectsofcomplexity,suchasthepriorvarianceofandinterdependence istobuildasequence(path)ofprobabilitydistributionsthatconnect
amongparameters.Bycontrast,thefreeenergyapproximationprovides thepriortotheposteriordistributionbyusingatemperatureparameter
amoreinformedmeasureofcomplexitythatisgenerallymoreappro- onthelikelihoodpartofthemodel(LartillotandPhilippe,2006).Inde-
priateforreal-worldbiologicalsystemswhichareimbuedwithparam- pendentsinglechainscanbeusedtoobtainsamplesfromeachdistribu-
eter interdependencies (Gutenkunst et al., 2007). However, tionfromthissequence;joiningthesamplesfrommultiplechainsyields
distributionalassumptionsmayhavegreaterimpactthanforBICand anasymptoticallyexactestimateofthelogevidence.Anadditionalim-
AIC(sincetheyconcernnotonlytheaccuracy,butalsothecomplexity provementistousepopulationMCMC(CalderheadandGirolami,2009)
term),andevaluatingthetightnessofitsboundapproximationrequires which imposes a dependency between neighbouring distributions
computationallyexpensivesamplingschemes(Aponteetal.,2016).On (chains) in thesequence andimproves the samples obtained from
theotherhand,thenegativefreeenergyapproximationwasshownto eachsinglechain.
exhibitbettermodelcomparisonperformancethanAIC/BICinthecon- Whilesampling-basedapproachestothelogevidenceareapromis-
textofregressionmodelsandDCM(Penny,2012)andalsoproved ingdirectingforfuturedevelopmentsofBMS,theyhaveusuallybeen
prohibitivelyexpensive(intermsofcomputetime)sofar.However,re-
centadvancesinexploitingthepowerofgraphicsprocessingunits
(GPUs) are now beginning to turn sampling approaches, including
5 Severalsimulationstudieshaveexaminedthevalidityoftheseapproximationsinthe
multi-chainandpopulationMCMCmethods,intoaviablealternative
contextofthemodelsdiscussedinthispaper;forexample,seeFigure4inPennyetal.
(2004a),Figures3–5inStephanetal.(2008),andFigures6–8inPenny(2012). forcomputingaccuratelogevidenceestimates(Aponteetal.,2016).