Download 1 Forthcoming in M. Massimi and J.W. Romeijn (eds.), European

ForthcominginM.MassimiandJ.W.Romeijn(eds.),EuropeanPhilosophyof
ScienceAssociation(EPSA15),Dordrecht:Springer.
Propensities,Probabilities,andExperimentalStatistics 1
5,169words
MauricioSuárez
DepartmentofLogicandPhilosophyofScience,
ComplutenseUniversityofMadrid,28040Madrid,Spain
Email:[email protected]
1. PluralismaboutObjectiveProbability
2. ReductiveAnalysesofChance
3. AgainsttheIdentityThesis
4. ChanceAssumptionsinStatisticalModelling
5. SavingtheStatisticalPhenomena
6. Conclusion
Abstract:Idefendathree-foldformofpluralismaboutchance,involvinga
tripartitedistinctionbetweenpropensities,probabilities,andfrequencies.The
argumenthasanegativeandapositivepart.Negatively,Iargueagainstthe
identitythesisthatinformscurrentpropensitytheories,whichalreadysuggests
theneedforatripartitedistinction.Positively,Iarguethatthatatripartite
distinctionisimplicitinmuchstatisticalpractice.Finally,Iapplyawell-known
1IthankaudiencesattheBSPS2015conferenceinManchesterandtheEPSA15
conferenceinDusseldorffortheircommentsandreactions.Thanksalsotothe
othermembersofthesymposiumpanelatEPSA15:LukeGlynn,AidanLyon,and
PhilipDawid,aswellastwoanonymousreferees.Researchtowardsthispaperwas
fundedbyaMarieCuriepersonalgrantfromtheEuropeanCommission(FP7PEOPLE-2012-IEF:Projectnumber329430),andresearchprojectFFI2014-57064PfromtheSpanishGovernment(MinistryofEconomicsandCompetitiveness).
1
frameworkinthemodellingliteratureinordertocharacterizethesethreeseparate
conceptsfunctionallyintermsoftheirrolesinmodellingpractice.
Keywords:chance,propensities,probability,statistics,modelling
1.PluralismaboutObjectiveProbability
RudolfCarnap(1945,1950)wasoneofthefirstanalyticalphilosophersof
sciencetoopenlydefendandpromotetheviewthatthereisnotjustonekindof
probabilitybutavarietyofkinds;andcorrespondinglynotjustone“probability”
concept,butapluralityofconcepts.Carnap’spluralismwasmodest:having
rejectedoneconcept,hesettledforthenextnumberup,namelytwoconcepts,so
minimizingthevarietyasmuchaspossible.Hecharacteristicallyreferredtothese
twoconceptsbymeansofindexes,asprobability1andprobability2.Probability1is
applicabletotheconfirmationoftheoriesbyempiricalevidence,andmore
particularlytotheconfirmationoftheoreticalsentencesbyso-calledprotocol
sentences.Thuswesaythataparticulartheoryismoreorlessprobableinthelight
ofevidence;andthatitismoreorlessprobablethansomecompetitorinthelight
ofsuchevidence;andwemayevenhavereasontoassertthatitsdegreeof
confirmation,orprobability,is0.9orsomeothersuchvalueintherealunit
interval.Thefirsttypeofprobabilityisthusnotamindorlanguageindependent
featureoftheworld.Itisratherafeatureorourtheoriesorlinguisticdescriptions
oftheworld.Inotherwordstheterm“probability1”belongsinwhatCarnapcalled
theformalmodeofspeech(Carnap,1935/37).
Thesecondkindofprobability,or“Probability2”,isbycontrastamindor
languageindependentobjectivefeatureoftheworld.Itdependsonthewaythe
worldisconstitutedandwhatthefactsare,regardlessofourlanguage,cognitiveor
mentalstates,beliefs,attitudesorabilities.Inotherwords,“probability2”isaterm
thatbelongsinthematerialmodeofspeech,andappearsinordinarydescriptions
oftheobjectiveprobabilitiesorchancesofparticularevents.Sciencemerely
extendsthisordinaryuseoflanguageinordertodescribeparticularphenomena
asstochasticbymeansofstatisticalorprobabilisticmodels.Thestatements
2
regarding“probabilities”thatappearinscientificmodels–inphysicsand
elsewhere–arethereforeallprimafacie“probability2”statements.
Carnapwentontoassociatethesestatementstostatisticalfrequenciesin
theempiricisttraditionofVonMises(1928)orReichenbach(1935).Wenowadays
thinkthatnomerestatisticalinterpretationof,say,thequantumstatevector,or
theprobabilitiesthatitentails,canbemadetowork.However,forCarnapthe
fortuneofafrequencyinterpretationofobjectiveprobabilityisamatterof
secondaryimportance–andhewasinfactacutelycriticalofsomekeyaspectsin
Reichenbach’sempiricistaccount.Carnap’smainconcernwasnottodefend
frequencies,butgenuineobjectiveprobabilities.Andwhilehisparticularcontrast
betweenlogicalandfrequencyconceptsofprobabilitydidnotperhapssucceed
well,theoveralltwo-foldpluralismdid.Thustwentyyearson,wefindIanHacking
(1975)drawingasimilartwo-folddistinctionbetweensubjectiveandobjective
aspectsofprobability.Morerecentworkinthephilosophyofprobability(e.g.
Gillies,2000)ifanythingentrenchesthiskindofpluralism,asapositivestateof
thingstobecelebrated.
2.ReductiveAnalysesofChance
Thephilosophyofobjectivechancehasthroughoutmuchofitshistory
pursuedareductionistagenda.Somephilosophershaveattemptedtoreduce
objectiveprobabilitiesorchancestofrequenciesorratiosin(virtualorreal)
sequencesofexperimentaloutcomes;othershaveattemptedtoreducethemto
propensities,understoodastheprobabilisticdispositionsofchancysetupsor
arrangements.2Suchreductiveexercisesareatleastprimafaciecontraryto
appearances.Considerafewstatementsofparadigmaticobjectivechancesas
expressedinthematerialmodeofspeech:
2LewisiananalysesofchanceinthespiritofHumemayberegardedasavarietyof
frequencyaccountsforthepurposesofthispaper.
3
1) Acoin’spropensitytolandheadswithacertainprobabilitywhentossed
asdisplayedinalongsequenceoftosses.
2) Smoking’spropensitytocauselungcancerwithacertainprobability,as
demonstratedbycontrolpopulationstatistics.
3) Thepropensityofaradioactiveatomtodecaywithacertainprobability
exhibitedinexperimentsrunonthematerial.
Thereareobviousdifferencesbetweenthecases.Thefirststatement
describesanordinaryoreverydaychance;thesecondoneinvolvesachanceto
causeaparticulareffect;andthefinalstatementreferstoaputativelyfundamental
andthereforeirreduciblechanceinatomicphysics.3Neverthelessallthese
statementsappearstoinvolvethreedistinctproperties:the“propensities”ofthe
chancyobject;the“certainprobabilities”thatsuchpropensitiesgiveriseto;and
the(finite,actual)frequenciesofthecorrespondingoutcomesobservedinan
experimentaltrialwhichdisplaysuchprobabilities.
Inotherwords,the“appearances”,asIshallcallthem,involvethreedistinct
properties.Yet,reductiveanalysesofchance(frequencyandpropensity
interpretationsofprobability)aimtoreducethemalltojustone,oratbesttwo.On
thefrequencyinterpretationpropensitiesareredundantandcanbediscarded
altogether;andprobabilitiescanbefullyanalysedintermsofeitherlongrun
actualfrequencies,orhypotheticallimitingfrequencies.Carnapreferstosuchan
identificationofprobabilitieswithfrequenciesasthe“identityconception”
(Carnap,1945,p.527).Andwhilethereisdebateamongstdifferentfrequency
schools,inparticularregardingthestatusandnatureofthelimitinghypothetical
frequencies,theyareallagreedontheessentialfactsaboutreduction.Onanyof
3Onemayinturnwonderwhetherallbonafidechancesultimatelyreduceto
physicalchances.Theanswerturnsonthethornyquestionofwhetherthe
“special”sciences,andindeedordinarycognitionofmacroscopicobjectsand
phenomena,ultimatelyreducetophysics.Iverymuchdoubtsuchreductionis
possibleordesirable,butmyclaimsinthispaperareindependentandrequire
neitherreductionismtophysicalchances,noritsdenial.
4
theseschoolsthereareonlyreallyfrequencies;everyreferencetoanyother
apparentconceptinthestatementsaboveisinfactredundant.
Ontheotherhandthepropensityinterpretationofprobabilitydefendedby
KarlPopper(Popper,1959)notoriouslyembracedasimilar(butincompatible)
identificationofprobabilitieswithpropensities,whichIhaveelsewherereferred
toasthe“identitythesis”(Suárez,2013).Onthisview,thereareofcoursefinite
frequenciesinactualexperimentalrunsofanyexperiment,buttheyneedhaveno
limitingproperties.Probabilitiesareatanyratenottobeidentifiedwitheitherthe
actualorthehypotheticallimitingfrequencies.Theyareinsteadpropensities.So,
onthisaccountthereareonlyreallyfrequenciesandpropensities;anyapparent
referenceinthestatementsaboveto“probability”asadistinctkindorpropertyis
ultimatelyredundant.
Eachofthesereductionshashadformidablechampionsthroughoutthe
historyofthesubject;infactbarelyanyphilosopherofprobabilityhasfailedto
attemptoneoranotherversionofthisreductionofchance.Yet,therearebynow
verystrongargumentsagainstbothkindsofreduction,whichsuggestthatthe
prospectsofareductionofprobabilityaredim.Ishallhereonlyreviewarguments
totheeffectthatprobabilitycannotbereducedtopropensity.Butthearguments
byAlanHajekandothersagainstfrequencyinterpretationsofprobabilityareat
leastasconvincing.4Allthreeconcepts(propensity,probability,frequency)seem
toberequiredforasatisfactoryunderstandingofobjectivechance.
MymainclaiminthisessayisthatwhatCarnapcalledprobability2isnotin
factamonolithicnotion.Ittooisplural,andcomposedofanarrayofthree
differentconceptsholdinginterestinglycomplexrelationstoeachother.In
additionIdoofcourseacceptsubjectiveprobabilitiesorcredences,andperhaps
alsodistinctlogicalorepistemologicalprobabilities(confirmatoryprobabilities).
Inotherwords,IverymuchshareCarnap’spragmaticpluralism,butwhereas
4SomeofHájek’sarguments(1997)relyonthewell-knownreferenceclass
problems.Iamnotsointerestedinthemherebecausetheyleaveopenanyclaim
regardingareductiontopropensities,andIamarguingforafulltripartite
distinction.
5
Carnaptriedtominimizethepluralismbyrestrictingittotwokindsofprobability,
Ifindgoodreasonsnowadaystowanttomaximizethepluralisminorderto
achieveafullunderstandingofobjectivechance.Therearebothnegativeand
positivereasonsformaximalpluralism.Thenegativereasonshavealltodowith
thefailuresofreductiveprogrammes(section3).Thepositivereasonsare
connectedwiththepresuppositionsofscientificpractice(section4)
3.AgainsttheIdentityThesis
Letmebrieflyreviewtheargumentfromthephilosophyofprobability
againsttheidentitythesisbetweenpropensitiesandprobabilities.5Theidentity
thesishastwoparts,orhalves,whichwemayrefertoasthepropensity-toprobabilityhalfandtheprobability-to-propensityhalf.Theformerassertsthatall
propensitiesare,orcanberepresentedas,probabilities.Thelatterstatesthatall
probabilitiesarepropensities,orcanbeinterpretedassuch.Togethertheymake
thefullclaimthatprobabilitiesandpropensitiesareextensionallyidentical.
Bothpartsoftheidentitythesisareinfactfalse,asisshownbydifferent
formsofwhatisknownasHumphreys’paradox.Thefalsityoftheprobability-topropensityhalfisatrivialconsequenceoftheasymmetriesofpropensities.Thisis
bestunderstoodbyconsideringacausalpropensitysuchassmoking’spropensity
tocauselungcancer(myexample2above).Supposeweestimateforaparticular
populationtheincidenceoflungcanceramongstsmokersat1%,whichwemay
writeasP(C/S)=0.01.Andsupposethatwealsohaveestimatesfortheprior
probabilitiesofsmokingandlungcanceracrossthepopulationat,say,20%and
0,5%respectively(P(S)=0.2andP(C)=0.005).Wemaytheneasilyestimatethe
inverseprobabilitybymeansofBayes’theorem:
P ( S C) =
P (C S ) P ( S ) 0.01 × 0.2
=
= 0.4 .
P (C )
0.005
5ThefullargumentmaybefoundinSuárez(2013,2014)ofwhichthissectionis
€
anelaborationandsummary.
6
Now,thefirsthalfoftheidentitythesis(theprobability-to-propensityhalf)
holdsthatprobabilitiesmaybeinterpretedaspropensities.Ifso,P(C/S)maybe
understoodasthepropensityofsmokingtocausecancer.Butitfollowsfromour
derivationthatP(S/C)isthenalsowell-definedat40%,soitmustalsoreceivea
propensityinterpretation,whichseemsjustimpossible:Thereissimplyno
propensityoflungcancertocausesmoking.
Whilethissimpletypeofargumentiswellknowntheconsequencesforthe
identitythesisarenotalwaysfullyappreciated.Propensitiesareasymmetricina
waythatprobabilitiesarenot.Theasymmetryisrevealedmoststrikinglyinthe
caseofcausalpropensities,butismoregenerallyafeatureofallpropensities
whetherornotcausal.Thusacoin’spropensitytolandheads,andaradioactive
atom’spropensitytodecayarealsoasymmetricinawaythatgeneratesasimilar
problemfortheirBayesinverseprobabilities.Wemayhopethattheotherhalfof
theidentitythesis(thepropensity-to-probabilityhalf)holdsnonetheless,andthat
itisstillthecasethatallpropensitiesareprobabilities.Whilethishalfonitsown
failsshortofafullreductiveanalysisforprobabilities,itisstillanaccountof
propensitiesasprobabilities.
However,PaulHumphreys’originalargumentdefeatsthishalfofthe
identitythesisaswell.Humphreys(1985)consideredathoughtexperimentwhere,
regardlessoftheoutcome,thepropensitiesofthesystemdescribedarenotand
cannotberepresentedasprobabilities.Heconsideredasourceemittingone
photonatatimet1,reachingahalfsilvermirrorattimet2,andbeingtransmittedat
timet3.Hethenplausiblystipulatedthatthefollowingthreeclaimsholdregarding
thepropensitiesofthephotoninthethoughtexperiment:
i)
Anyphotonthatreachesthehalfsilvermirrorhassomefinite(nonzero)propensitytobetransmitted.
ii)
Anyphotonthatisemittedhassomepropensitygreaterthanzerobut
notonetoreachthemirror.
7
iii)
Anyphotonthatisemittedandfailstoreachthemirrorhaspropensity
zero(i.e.ithasnopropensity)tobetransmitted.
Theseclaimsmayallberegardedasuncontroversialregardingthethought
experimentathand.Theyallcertainlyseemveryplausible.Humphreysrendered
theseclaimsinaconditionalprobabilityformulationthatishoweverfarfrom
innocuousorobvious,asfollows:
i) Pt 3 (Tt 3 It 2 & Bt1 ) = p > 0 .
ii) 1 > Pt1 ( It 2 Bt1 ) = q > 0 .
€
iii) Pt1 (Tt 3 ¬It 2 & Bt1 ) = 0 .
€
€
Eachoftheseformalconditionsismeanttocapturefullyeachofthe
correspondingphysicalclaimsregardingthepropensitiesatworkinthethought
experiment.Thisassumesthatthereisalwaysauniquerepresentationfor
propensitiesintermsofconditionalprobabilities.Yet,thesethreeformal
conditionsareinconsistentwiththeKolmogorovaxioms,andinparticularwiththe
fourthaxiomforconditionalprobability(alsoknownastheratioanalysisof
conditionalprobability).
Now,thereareanumberofcaveatstoHumphreys’proof,whichIcannot
discusshereinfull,butdeserveabriefmention.Firstly,theproofassumesa
principleofconditionalindependencewherebypropensitiesdonotactbackwards
intime: Pt1 ( It 2 Tt 3 & Bt1 ) = Pt1 ( It 2 ¬Tt 3 & Bt1 ) = Pt1 ( It 2 Bt1 ) .Theprincipleisindeed
questionableingeneral,sincethereisnoreasonwhypropensitiesshouldbeany
moreforward-lookingthancauses.Inotherwords,theasymmetryofpropensities
€
isnottheasymmetryoftime,butisratherasuigenerisasymmetry,whichmayor
notcoincidewithtemporalasymmetry.Yet,inthethoughtexperimentathand,the
applicationofconditionalindependenceislegitimate–inotherwordsthe
propensitiesthatobtaininthethoughtexperimentareallasamatteroffact
forwardlooking.Sotheproofdoesnothangonthisassumptionbeinggenerally
valid.
8
Thesecondcaveatisthatrelinquishingtheratioanalysisofconditional
probability(i.e.givingupontheKolmogorovcalculus,oratleastontheimplicated
fourthaxiom: P ( A B) =
P ( B A)
P ( B)
),doesnotactuallydeliverusfromcontradiction,
andcannotinfactgetusoutoftrouble.Theratioanalysisisindeeda
presuppositionoftheKolmogorovcalculus,butnoothercalculithatwehavesofar
€
developedisinabetterpositiontoovercomeHumphreys’paradox.6
Tosumup,Humphreys‘proofisrightlywidelyunderstoodtoshowthatthe
representationofpropensitiesintermsofconditionalKolmogorovprobabilitiesis
flawed:Propensitiesarenotingeneralprobabilities.
4.ChanceAssumptionsinStatisticalModelling
Thesecondandmainargumentforpluralismdoesnotrelyonformalissues
intheaxiomatizationofprobability.Itisratherrelatedtothepracticeofstatistical
modelling.Ishallarguethatthe“appearances”(i.e.thetripartitedistinction
betweenpropensities,probabilities,andfrequencies)aretacitlypresupposedin
muchofthismodellingpractice.Andphilosophersofscienceasithappensarewell
equippedtounderstandthetripartitedistinctionaspartofanymodellingpractice
–sinceitfollowsfromaninfluentialaccountofmodellingingeneral.
Astatisticalmodelisoftenpresentedasapairstructure,consistingofa
sampleoroutcomespaceSandasetofprobabilitydistributions,ordistribution
functionsPi,definedoverthissample: 〈S,Pi 〉 .Thedomainofeachoftheprobability
functionsisasubsetorpowersetoftheelementsinthesampleandtherangeof
eachprobabilityfunctionisofcoursetheunitrealnumberinterval.7However,the
€
6ForaverynicetreatmentofthisissueinconnectionwithRenyi’saxiomsystem,
seeLyon(2013).
7Theliteratureonstatisticalmodellingislarge.Myunderstandingisinformed
mainlybyCox(2006),Freedman(2009),andKrzanowski(1998),inadditionto
theMcCullaghpaperdiscussedinthetext.
9
simpledefinitionhasburiedwithinitafairamountoftacitstructurethatisrarely
madeexplicit.Inparticulartheselectionofthesampleordomainofastatistical
modelisnotatrivialmatterandinvolvesconsiderablejudgement.Inhisinfluential
(2002)paper,PeterMcCullaghshowshowanystatisticalmodelofastochastic
phenomenoninvolvestwostrictlydistinctdomains:thedomainofthe
phenomenoninquestion,andthedomainoftheprobabilityfunctionscontainedin
themodel.Theideaisthatthephenomenonisfirstdescribedasasetof
parametersΘ,inwhatmaybecalledaprepareddescription.Astatisticalmodelis
thenafunctionthatmapseachparameterpointinΘontooneoftheprobability
functions℘( S ) definedoverthesamplespace.Inotherwordsastatisticalmodelis
functionallyamap: P : Θ →℘( S ) whichassignstoeverypoint θ t ∈ Θinthe
parametersetthatrepresentsthephenomenonacorrespondingprobability
€
function℘θ i ( S ) definedoverthesamplespace.AsMcCullagh(2002,p.1225)
€
€
notes:“itisimportanttodistinguishbetweenthemodelasafunction P : Θ →℘( S ) ,
andtheassociatedsetofdistributions℘θ i ( S ) ⊂℘( S ) ”.
€
€
Astatisticalmodelcomprisesboththeparametersetandthesetof
€
probabilityfunctionsoverthesamplespace.So,implicitly,astatisticalmodelis
definedovertwodistinctdomains:ΘandS.Theformerdomainappearsmerelyas
asubscripttotheprobabilitydistributionfunction.Itisthelatterdomain,the
sampleoroutcomespace,thatisthepropersigmafieldoverwhichthe
probabilitiesaredefined.Itfollowsthenthattheprobabilitiesinastatisticalmodel
arenotdefinedovertheparametersetthatrepresentsthephenomenonin
question.Therelationshipbetweenthesamplespaceandtheparametersetis
ratherindirect;andthemostimportantquestionforanymodellerispreciselyhow
to‘convert’theparameterspacemeaningfullyintothesampleoroutcomespace.
Thereisnotrivialalgorithmicprocedure:Itisratherahighlycontextualmatterof
judgement,relativetotheparticularproblemathand.Iteffectivelytransformsa
questionregardingthephenomenonanditscausesintoaquestionregardingthe
probabilitiesinthemodel.Themodelmustofcoursebeconsistentwithknown
datarelativetothephenomenon,butitishardtoseehowitwouldnotinvolve
idealizationofonesortofanother.Thisisafterallonemoreinstanceof‘modelling’
10
thephenomenainastreamlineddescription.Toquotefromthedistinguished
statisticianDavidCox(2006,p.197):
“Formalizationoftheresearchquestionasbeingconcernedwith
aspectsofaspecifiedkindofprobabilitymodelisclearlyofcritical
importance.Ittranslatesasubject-matterquestionintoaformalstatistical
questionandthattranslationmustbereasonablyfaithfuland,asfaras
feasible,theconsistencyofthemodelwiththedatamustbechecked.How
thistranslationfromsubject-matterproblemtostatisticalmodelisdoneis
oftenthemostcriticalpartofananalysis.Furthermore,allformal
representationsoftheprocessofanalysisanditsjustificationareatbest
idealizedmodelsofanoftencomplexchainofargument.”
Themostimportantconstraintinstatisticalmodellingisthis:The
derivationofthesamplespacefromtheparameterspacemustberesponsiveto
thefeaturesofthephenomenon.ThefunctionthattakesfromtheparametersetΘ
totheprobabilitydistribution℘θ i ( S ) isnotarbitrary,butdependssensitivelyupon
thenatureofthephenomenoninquestion.Inparticular,ifthephenomenonis
dynamicalthefunctionmustrespondtoitsdynamicallaws.Andifthelawsare
€
stochastic,thenitmustrespondtotheobjectivechancesthatappearinthoselaws.
8Sothefunctionthatyieldsthesampleoroutcomespaceofanystatisticalmodelof
anindeterministicphenomenonalreadyhasbuiltintoitadependenceuponsome
antecedentchances,whichappearinthephenomenonasdescribed.Themodelis
thenofcoursetestedagainstexperimentsrunuponthesystemandyieldingfinite
frequencies.Itthenfollowsthatatripartitedistinctionbetweenpropensities,
8Considerasarudimentaryexampletwofaircoins,eachindependentlyobeyinga
binomialdistribution.Supposethatthecoinsarethenphysicallyconnectedin
accordancetoadynamicallawthatimpliescorrelationsamongstthem(youcan
imaginesomekindofinvisiblethreadconnectingbothtailsides).Theyare
thereafteralwaystossedsimultaneouslyandmorelikelytofallonthesameside.
Thesamplespaceinthestatisticalmodelforthisphenomenonmusttheninclude
bothoutcomeevents(“head”and“tails”)foreachofthecoins,aswellasallthe
jointevents(“heads&heads”“heads&tails”,etc).Andtheprobabilitydistribution
functiondefinedinthisformalmodelmustbeconsistentwiththeseunderlying
dynamicalfacts.
11
probabilities,andfrequenciesisalreadyimplicitlyassumedinthepracticeof
statisticalmodelling.9
5.SavingtheStatisticalPhenomena
Thestatisticalmodellingofaphenomenonthuspresupposesathree-fold
distinctionbetweenthepropensitiesresponsibleforthephenomenon,the
probabilitiesthatappearinthestatisticalmodelofthephenomenon,andthe
(actual,finite)frequenciesintheexperimentalsequencesthattestthemodel.This
tripartitedistinctioninstatisticalmodellingpracticeisanaturalconsequenceof
theapplicationtostochasticphenomenaofacorrespondingtripartitedistinction
inmodellingmoregenerally:BogenandWoodward’s(1988)distinctionbetween
theory,phenomena,anddata.10Ontheiraccount,whichisbackedupbyanumber
ofdetailedcasesstudies,themainroleofatheoryistoexplainaphenomenon;and
whiletheoriescanbeconsistentwithdata,theyarenotinthebusinessof
explainingorpredictingdata.Correlatively,anexperimenttypicallyyields
observabledata(thatis:finite,actualrecordsofparticularobservationsor
measurementoutcomes);butthephenomenonitselfcannotbesoobserved.
Ratherourknowledgeofaphenomenonistheresultofanumberoflow-level
inferencesthatestablishaparticularmodelforit.Theinferredphenomenonis
thendescribedinthemodelandwithluckexplainedbyatheory.Itshouldbeclear
thattheaccountpresupposesthatthereareindependentfunctionalrolesforeach
ofthethreecomponents:theories,phenomena,anddata.
Thebest-knownillustrationofthetripartitemodelthatBogenand
Woodwardprovideuswithistheoldexampleofthemeltingtemperatureoflead.
9ArefereehelpfullypointsoutthatSpanos(2006)defendsasimilardistinction
betweenstructuraltheorymodels,statisticalmodels,andobservationaldata,with
similarconsequencesregardingtheroleof“chanceset-ups”.
10Idonottherebyendorsehereanyoftheirepistemologicalclaimsbeyondthe
tripartitedistinction.Myaccountofstatisticalmodelling,forinstance,isalso
consistent–atleastforthepurposesofthepresentessay–withthewidely
acceptedclaimthatmodelsareautonomousrelativetoboththeoryanddata.See
theessaysinMorrisonandMorgan(1999)foranarticulationanddefence.
12
11Idonotneedtoreviewtheirdiscussioningreatdetailformypurposeshere.Itis
enoughtoemphasisethatthefactthatleadmeltsat327.5degreesCelsiusisnota
pieceofobservabledataontheiraccount,butmayonlybeinferredfromavery
complexarrayofdatabysomesophisticateddataanalysis.Thedatapointsthatare
infactobservedcorrespondtosinglerecordingsofmeasurementstakenon
particularsamplesofleadunderveryparticularconditions–andhencesubjectto
hugevariationinexperimentaland/orsystematicerror.Thevariationissolarge
infactthattheremayexistnodatapointcorrespondingtotheprecisemelting
temperatureoflead.And,contrarytowhatthelogicalpositiviststhought,no
theoryinsolid-statephysicsmaybeabletoexplainasingledatapoint.
Thetheoryofphasetransitionsformetalsindeedexplainsthedifferent
meltingtemperaturesofthedifferentmetalsintermsoftheirintermolecular
forces,anditpredictsthecriticalenergyrequiredtoovercomethemolecular
bondstypicalofacrystallinesolid.Butonthetripartiteaccount,itdoesnotneedto
predictorexplaineachoranyparticularmeasurementrecord.Butthenthetheory
isnotintendedtoeverdothat.Itisrathermeanttoaccountfor,andexplain,the
onetrueclaimaboutthephenomenon,i.e.theinferredfactthatleadmelts
preciselyat327,5degreesCelsius.
Inotherwords,onthisaccountofmodellingpractice,thereisaclear-cut
functionaldistinctionbetweentheories,modelsofphenomena,andobservable
data.Thetheoriesareinthebusinessofexplainingandpredictingphenomena,not
data.Thephenomenaaredescribedbymeansofmodelsthatareinturninferred
fromthedatabycomplexstatisticalanalyses.Theobservabledataareusedto
confirmthesemodelsofphenomenabuttheycannotdirectlybeemployedinthe
confirmation(orrefutation)oftheory.Nofinitenumberofcontradictingdata
pointscanrefuteatheory,especiallywhenthedatatakentogethervindicatea
11TheyderivetheexamplefromErnstNagel’s(1961)discussion.OneofBogenand
Woodward’smainclaimsisthatthelogicalpositivistaccountsofexplanationand
confirmationsuffersfromoversimplificationoftheempiricalcontentofscience.
Thelogicalpositivistemphasison“observablephenomena”is,accordingtoBogen
andWoodward,anoxymoron.Asexplainedinthetext,phenomenaareontheir
accountneverobservable,butalwaystheresultofsomelowlevelgeneralizing
inferences.
13
phenomenonthatiscompatiblewiththetheory.Inthistripartiteaccountthe
connectionsbetweendataandtheoryarealwaysmediatedby(amodelof)
phenomena.
Statisticalmodellingisonetypeofscientificmodellingpractice.Soitstands
toreasonthatitshouldexhibitthesamefunctionaldistinctionsthatareoperative
inmodellingingeneral.Andindeeditdoes.Statisticiansdrawtherelevant
distinctions,particularlywhentheyreflectupontheirpractice.Inparticular,as
shownintheprevioussection,theydistinguishcarefullytheformal“statistical
model”pair 〈S,Pi 〉 fromthephenomenonitself,howeverparametrizedundersome
theoreticaldescription.Withsomeingenuityatheorymaybefoundthatdescribes
thedynamicalpropertiesbehindthephenomenon,includingitspropensities.Ifso,
€
anexplanationisthusprovidedforthephenomenonasdescribedinaformal
statisticalmodel–bymeansofasetofprobabilitydistributionfunctionsdefined
uponanappropriatesamplespace.Thesemodelsareinturntestedagainstthe
data–namelythefrequencyratiosrevealedinlongbutfinitesequencesof
experimentaloutcomes.12
Radioactivedecayratesareagoodillustration.Anatom’schanceor
probabilityofdecayisapropensityofthematerial,asdescribedbyatomictheory.
Thetheoryinvokessuchpowersaspartoftheexplanationofthetypicalrateof
decay(halflife)ofthematerial.Itdoesnottherebyexplainanyparticularatom’s
eventofdecay(orotherwise).Thisisanindeterministicsingleeventthatcannot
besoexplained.Andnoparticulareventofdecay(ornon-decay)canonitsown
provideanyconfirmationorrefutationfororagainstatomictheory.Theonlyway
datacanpossiblyimpingeontheoryisindirectlyviatheprobabilitydistribution
12Inourrudimentarytwo-coinsystemexample,thetheorythatdescribesthe
dynamicsofthesystem(includingthehiddenmechanism,suchastheconnecting
thread)isnotmeanttoaccountfor,orexplain,anyparticulartwo-coinoutcome.It
isonlymeanttoexplaintheprobabilitydistributionthatappearsintheformal
statisticalmodelforthephenomenon.Similarlynoparticularoutcomemayrefute
thistheoryotherthanbycompromisingthedistributionfunctioninthemodel–
forwhichmuchmorethanjustoneobservationwillcertainlybeneeded.
14
functionsinthestatisticalmodelthataccountsforthephenomenaofradioactive
decay.13
7. Conclusion
Ihavearguedformaximalpluralismaboutchance,byprovidingnegative
andpositiveargumentsforatripartitedistinctionbetweenpropensities,
probabilities,andfrequencies.Humphreys’paradoxprovidesgroundsforthe
distinction,sinceitmakesitveryimplausiblethatchancesmaybereducedor
analysedawayinanyfewerterms.Inaddition,Ihavepositivelydisplayed
elementsinthepracticeofstatisticalmodellingrecommendingthesame
distinction.Finally,Ihavearguedthatthetripartitedistinctionmakesfullsense
withinaninfluentialcurrentaccountofmodellingpractice.
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