Selectivity in Probabilistic Causality: Where Psychology Runs Into Quantum Physics

0.16 MB·English

by  Ehtibar N. Dzhafarov

Selectivity inProbabilisticCausality:
WherePsychologyRuns Into Quantum Physics
Ehtibar N. Dzhafarov∗
Purdue University
Janne V. Kujala
University of Jyväskylä
2 Givenasetofseveralinputsintoasystem(e.g.,independentvariablescharacterizingstimuli)andasetofseveralstochasticallynon-independent
1
outputs(e.g.,randomvariablesdescribingdifferentaspectsofresponses), howcanonedetermine, foreachoftheoutputs, whichoftheinputsit
0
isinfluenced by? Theproblem hasapplications ranging frommodeling pairwisecomparisons toreconstructing mental processing architectures
2
toconjointtesting. AnecessaryandsufficientconditionforagivenpatternofselectiveinfluencesisprovidedbytheJointDistributionCriterion,

n accordingtowhichtheproblemof“whatinfluenceswhat”isequivalenttothatoftheexistenceofajointdistributionforacertainsetofrandom
a variables. Forinputsandoutputswithfinitesetsofvaluesthiscriteriontranslatesintoatestofconsistencyofacertainsystemoflinearequations
J andinequalities(LinearFeasibilityTest)whichcanbeperformedbymeansoflinearprogramming. Whilenewinthebehavioralcontext,boththis

3 testandtheJointDistributionCriteriononwhichitisbasedhavebeenpreviouslyproposedinquantumphysics,indealingwithgeneralizationsof
1 Bellinequalitiesforthequantumentanglementproblem. Theparallelsbetweenthisproblemandthatofselectiveinfluencesinbehavioralsciences
areestablishedbyobservingthatnoncommutingmeasurementsinquantumphysicsaremutuallyexclusiveandcanthereforebetreatedasdifferent

] levelsofoneandthesamefactor.
n
a
- KEYWORDS:Bell-typeinequalities,EPRparadigm,factorialdesign,Fine’sinequalities,jointdistributioncriterion,probabilisticcausality,mental
a architectures,randomoutputs,selectiveinfluences,quantumentanglement,Thurstonianscaling.
t
a
d
. 1. INTRODUCTION pendenceofthejointdistributionsoftherandomoutputsA,B,C
s ontheexternalfactorsa ,b ,g ,d ?
c
si This paper deals with diagrams of selective influences, like These questions are important, because the assumption of
y thisone: stochastic independence of the outputs more often than not is
h
eitherdemonstrablyfalseoradoptedforexpediencyalone,with
v4 [p a (cid:15)(cid:15) (cid:0)(cid:0)✁uu❥◆✁◆❥✁◆✁❥✁◆❥✁◆✁❥✁◆❥◆b❥(cid:15)(cid:15)◆❥◆❥◆❥◆❥◆❥◆❥◆❥'' g❥(cid:15)(cid:15)(cid:15)(cid:15)❥(cid:0)(cid:0)✁❥✁❥✁✁❥✁❥✁✁❥✁d (1) ntuliaobvniigoqtyutuhaiientgorecujausousfistnitafihltehcraeemtoliaorotendito.einclAsas.ltbFmtehotoewrdseieanelmsnintaeignnt,pciomeuf,tetseintanhnTebdheasiusntsrogsuctmbhouanpsietlti’tioscinnmooutohftspestuegvtleesenrciys--
A B C
8
eral modelof pairwise comparisons(Thurstone, 1927)it is as-
8
sumedthateachofthetwostimuliismappedinto“its”internal
3
The Greek letters in this diagram represent inputs, or external representation, while the two representationsare stochastically
2
. factors, e.g., parameters of stimuli whose values can be cho- interdependentrandomentities.InDzhafarov(2003),Dzhafarov
0
senatwillorobservedandrecorded. ThecapitalRomanletters and Gluhovsky (2006), and Kujala and Dzhafarov (2008) the
1
standforrandomoutputscharacterizingreactionsofthesystem readermay find other motivatingapplicationsfor the notion of
1
1 (an observer, a group of observers, stock market, a set of pho- selectiveinfluences: same-differentcomparisons,conjointtest-
: tons,etc.). Thearrowsshowwhichfactorinfluenceswhichran- ing,parallel-serialnetworksofmentaloperations,responsetime
v
dom output. The factors are treated as deterministic entities: decompositions,andallconceivablecombinationsofregression
i
X even if a ,b ,g ,d in reality vary randomly(e.g., beingrandomly analysisandfactoranalysis. Inthispaperweaddanothermoti-
r generated by a computer program, or being concomitant pa- vatingexample,thequantumentanglementprobleminquantum
a rameters of observations, such as age of respondents), for the physics.
purposes of analyzing selective influences the random outputs
Thispapercontinuesandexpandstheanalysisofselectivein-
A,B,Carealwaysviewedasconditioneduponvariouscombina-
fluencespresentedinDzhafarovandKujala(2010). Thefamil-
tionsofspecificvaluesofa ,b ,g ,d . Thefirstquestiontoaskis:
iaritywithitcanbehelpful,butthemainconcepts,terminology,
whatisthemeaningoftheabovediagramiftherandomoutputs
and notation are recapitulated in Section 2. Unlike in Dzha-
A,B,C in it are not necessarily stochastically independent? (If
farov and Kujala (2010), however, here we do not pursue the
theyare,theanswerisofcoursetrivial.) Andoncethemeaning
goalofmaximalgeneralityofformulations,focusinginsteadon
of the diagram of selective influences is established, how can
theconceptualset-upthatappliestocommonlyencounteredex-
one determine that this diagram correctly characterizes the de-
perimentaldesigns. Thismeansafinitenumberoffactors,each
havingafinitenumberofvalues. Italsomeansthattherandom
outcomes influenced by these factors are random variables in
the narrow sense of the word: their values are vectors of real
∗Corresponding author: EhtibarDzhafarov, PurdueUniversity, Department of
numbers or elements of countable sets, rather than more com-
PsychologicalSciences,703ThirdStreetWestLafayette,IN47907,USA.email:
[email protected]. plexstructures,suchasfunctionsorsets. Thisisdoneprimarily
1
2 DzhafarovandKujala
tosimplifyandshortenexposition,andalsobecausetheLinear 2.1. Factors,factorpoints,treatments
Feasibility Test, a new (for behavioral sciences) application of
theJointDistributionCriteriononwhichwefocusinthispaper A factor a is treatedas a setof factorpoints, eachof which
(Section3),isconfinedtofinitesetsoffinite-valuedfactorsand hastheformat“value(orlevel)x offactora .” Insymbols,this
finite-valuedrandomvariables.Thisalsoallowsustoemphasize can be presented as (x,‘a ’), where ‘a ’ is the unique name of
a simple but important and previously overlookedproposition, the set a rather than the set itself. It is convenientto write xa
Theorem2.3,whichessentiallysaysthat,whendealingwithob- in place of (x,‘a ’). Thus, if a factor with the name ‘intensity’
servablerandomvariables,theunobservablerandomentitiesof hasthreelevels,‘low,’‘medium,’and‘high,’thenthisfactoris
the theory can also be assumed to be random variables (in the takentobetheset
narrow sense). In another respect, however, the present treat-
mentismoregeneralthanthatinDzhafarovandKujala(2010): intensity= lowintensity,mediumintensity,highintensity .
we allow for incomplete designs, those in which some but not
necessarilyallcombinationsofthevaluesofthefactorsserveas There is no ci(cid:8)rcularity here, for, say, the factor(cid:9) point
allowabletreatments. Thismodificationiscriticalforthepossi- lowintensity stands for (value=low,name=‘intensity’) rather
bility of representingany diagramof selective influences, such than(value=low,set=intensity).
as(1),inacanonicalform,witheveryrandomoutputbeingse- WewilldealwithfinitesetsoffactorsF ={a 1,...,a m},with
lectivelyinfluencedbyoneandonlyonefactor. eachfactora ∈F consistingofafinitenumberoffactorpoints,
As it turns out, both the Linear Feasibility Test and the a = va ,...,va .
JointDistributionCriteriononwhichitisbasedhavetheirana- 1 ka
loguesinquantumphysics.1 Toappreciatetheanalogy,however, Clearly,a ∩b =∅forany(cid:8)distincta ,b (cid:9)∈F .
one has to adopt the interpretation of noncommuting quantum A treatment, as usual, is defined as the set of factor points
measurementsperformedona givencomponentofa quantum- containingonefactorpointfromeachfactor,
entangledsystemasmutuallyexclusivefactorlevelsofthesame
factor. In Sections 2.6 and 3 we discuss the parallels between f = xa11,...,xamm ∈a 1×...×a m.
theexistenceofaclassicalexplanationforanentanglementsit-
uationinquantummechanicsandtheadherenceofabehavioral The set of treat(cid:8)ments (used(cid:9)in an experiment or considered
experimenttoadiagramofselectiveinfluences. in a theory) is denoted by T ⊂a 1×...×a m and assumed to
benonempty. NotethatT neednotincludeallpossiblecombi-
The term “test” in this paper is used in the meaning of nec-
nations of factor points. This is an important consideration in
essary (sometimesnecessaryand sufficient) conditionsfor dia-
view of the “canonicalrearrangement”described below. Also,
gramsofselectiveinfluences.Theusageisthesameaswhenwe
incompletelycrosseddesignsoccurbroadly—inanexperiment
speakofthetestsforconvergenceincalculusorfordivisibility becausetheentireseta ×...×a maybetoolarge,orinathe-
1 m
inarithmetic. Thatis,themeaningofthetermisnon-statistical.
orybecausecertaincombinationsoffactorpointsmaybephysi-
We assume that random outputs are known on the population
callyorlogicallyimpossible(e.g.,contrastandshapecannotbe
level. Generalconsiderationsrelatedtostatisticaltestsbasedon
completelycrossedifzeroisoneofthevaluesforcontrast).
ourpopulationleveltestsarediscussedin Section3.6, butspe-
cificstatisticalissuesareoutsidethescopeofthispaper.
2.2. Randomvariables
Weassumethereaderisfamiliarwiththenotionofarandom
entity (randomvariable in the generalsense of the term) A as-
2. BASICNOTIONS sociatedwithanobservationspace(A,S ),whereA isthesetof
possible valuesforA, and S a sigma-algebra(setof events) on
A. Arandomvariable(inthenarrowsense)isaspecialcaseof
Inthissection,weestablishtheterminology,notation,andre-
arandomentity,definedasfollows:
capitulate basic facts related to factors, random variables, and
(i) if A is countable, S is the power set of A, then A is a
the dependence of the latter on the former. We follow Dzha-
randomvariable;
farovandKujala(2010),addingobservationsrelatedtothefac-
(ii)ifA isanintervalofreals,S istheLebesguesigma-algebra
torialdesignsbeing incompleteandrandomoutputsbeing ran-
onA,thenAisarandomvariable;
domvariablesinthenarrowsenseoftheterm. Attheendofthe
(iii) if A ,...,A are random variables, then any jointly dis-
section we discuss the parallels between the issue of selective 1 n
tributedvector(A ,...,A )whoseobservationspaceisthecon-
influenceinbehavioralsciencesandthequantumentanglement 1 n
ventionally understood product of the observations spaces for
problem.
A ,...,A isarandomvariable.
1 n
We use the relational symbol ∼ in the meaning of “is dis-
tributedas.” A∼BiswelldefinedirrespectiveofwhetherAand
Barejointlydistributed.
1WearegratefultoJeromeBusemeyerofIndianaUniversitywhopointedout Let,foreachtreatmentf ∈T,therebeavectorofjointlydis-
tousthattheformulationoftheJointDistributionCriterioninourearlierwork
tributedrandomvariablesA=(A ,...,A )withafixed(product)
hasthesameformalstructureastheidenticallytitledcriterioninFine(1981a- 1 n
b),inhisanalysisofquantumentanglement. observationspace and the probabilitymeasure µf that depends
SelectivityinProbabilisitcCausality 3
onf .2 Thenwesaythatwe haveavectorofjointlydistributed We can simplify the subsequent discussion without sacrificing
randomvariablesthatdependsontreatmentf ,andwrite generalitybyagreeingtoreduceeacharrowdiagram(inthecon-
textofselectiveinfluences)toabijectiveform,byappropriately
A(f )=(A1,...,An)(f ), f ∈T. redefiningfactorsandtreatments. Itisobvioushowthisshould
bedone. GiventhesubsetsoffactorsF ...,F determinedby
A correct way of thinking of A(f ) is that it represents a set of anarrowdiagram(2),eachF canbevie1wedasnafactoridenti-
vectors of jointly distributed random variables, each of these i
fiedwiththesetoffactorpoints
vectorsbeing labeled (indexed)by a particulartreatment. Any
subvector of A(f ) should also be written with the argumentf ,
say,(A1,A2,A3)(f ). Iff isexplicatedasf = xa11,...,xamm ,we a ∗i = (f F i)a ∗i :f ∈T ,
writeA(f )=A(xa 1,...,xa m). n o
It is important1to notemthat for distinct tre(cid:8)atments f an(cid:9)d f in accordance with the notation we have adopted for factor
the correspondingA(f 1) and A(f 2) do not possess a j1oint dis2- ptoooin,tasn:d(fthF ei)faa∗ic=tor(fa F∗i,c‘oa n∗s’i)s.tsIfofF oinislyemthpetdy,utmhemnyf fFaictiosrempopitnyt
tribution, they are stochasticallyunrelated. This is easy to un- i
derstand: sincef andf aremutuallyexclusiveconditionsfor ∅a i (where ∅ denotesthe empty set). The set of treatmentsT
observing values1of A, th2ere is no non-arbitraryway of choos- fortheoriginalfactors{a 1,...,a m}shouldthenberedefinedfor
ingwhichvaluea=(a1,...,an)observedatf 1shouldbepaired thevectorofnewfactors(a ∗1,...,a ∗n)as
with whichvalue a′ =(a′,...,a′) observedat f . To consider
A(f 1) and A(f 2) stochas1tically nindependent and2to pair every T∗= (f F 1)a ∗1,...,(f F n)a ∗n :f ∈T ⊂a ∗1×...×a ∗n.
possible value of A(f ) with every possible value A(f ) is as
1 2 nn o o
arbitrary as, say, to consider them positively correlated and to We call this (re)definition of factor points, factors, and treat-
paireveryquantileofA(f 1)withthecorrespondingquantileof mentsthecanonical(re)arrangement. We cansaythattheran-
A(f 2). dom variables following canonical (re)arrangement can be in-
dexed by the corresponding factors. Thus, when convenient,
2.3. Arrowdiagrams,canonically(re)arranged we{txeca.ic}aTn⊂hwef r.niAtoetsaitunios(un3a)fl,AFw{iae=1w}fri{intaeip}xlaatchieeinnopfilnAadc1ie,caoAtfe{as{2xt}ahiie}n:spinlagcleetoonf Ase2t,
GivenasetoffactorsF ={a ,...,a }andavectorA(f )=
1 m
(A1,...,An)(f )ofrandomvariablesdependingontreatment,an f {a i}= xa11,...,xann {a i}=xaii.
arrowdiagramisamapping
(cid:8) (cid:9)
F
M:{1,...,n}→2 (2)
2.4. Thecriterion
(2F beingthesetofsubsetsofF ). Later, inDefinition2.1, the
arrowswillbeinterpretedasindicatingselectiveinfluences,but Definition 2.1 (Selective influences, bijective form). An arrow
fornowthisisunimportant.Theset diagram(3)issaidtobethediagramofselectiveinfluencesfor
(A ,...,A )(f )and(a ,...,a ),andwewrite
1 n 1 n
F =M(i), (i=1,...,n),
i
(A ,...,A )"(a ,...,a ),
1 n 1 n
isreferredtoasthesubsetoffactorscorrespondingtoA. Itde-
i
termines,foranytreatmentf ∈T,thesubtreatmentsf F i defined if, for some random entity R and for any treatment f =
as xa 1,...,xa n ∈T,
1 n
f F i ={xa ∈f :a ∈F i}, i=1,...,n. (cid:8) (A1,..(cid:9).,An)(f )∼ f1(f {a 1},R),...,fn(f {a n},R)
Subtreatmentsf F acrossallf ∈T canbeviewedasadmissible (4)
valuesofthesubsietoffactorsF i (i=1,...,n). Notethatf F i is (cid:0)= f1(xa11,R),...,fn(xann,R) (cid:1),
emptywheneverF isempty.
i
The simplest arrow diagram is bijective, with correspon- where fi:a i×R →Ai(i=1(cid:0),...,n)aresomefunction(cid:1)s,withR
dences denotingthesetofpossiblevaluesofR.3
a 1 ... a n . (3) This definition is difficult to put to work, as it refers to an
existenceofa randomentity(variable)Rwithoutshowinghow
(cid:15)(cid:15) (cid:15)(cid:15) onecan findit or provethatitcannotbe found. The following
A1 ... An criterion(necessaryandsufficientcondition)for(A1,...,An)"
(a ,...,a )circumventsthisproblem.
1 n
2Theconvenient assumptionoftheinvariance oftheobservationspaceforA
withrespecttof isinnocuous:onecanalwaysredefinetheobservationspaces 3It will be shown below, Theorem 2.3, that random entity Rcan always be
fordifferenttreatmentsf tomakethemcoincide. chosentobearandomvariable(inthenarrowsense).
4 DzhafarovandKujala
Criterion 2.2 (Joint Distribution Criterion, JDC). A vector of Proof. The first statement follows from the fact that R can be
randomvariablesA(f )=(A ,...,A )(f )satisfiesadiagramof chosentocoincidewiththeJDC-vectorH,sothat
1 n
selective influences(3)ifandonlyifthereisavectorofjointly
a a
distributedrandomvariables fi(x i,H)=Hxaii,
fora 1 fora n fori=1,...,n,andxa i∈a i.TheJDC-vectorHisarandomvari-
H=Hxa11,...,Hxak11,...,Hxa1n,...,Hxaknn, afabrloev. &ThGelusehcoovnsdkys,ta2t0e0m6e,nbtasfeodlloownsafgreonmerTalhreeosruelmtfo1risntaDndzahrad-
z }| { z }| {
  Borelspaces(e.g.,inKechris,1995,p. 116).
one randomvariable for each factor pointof each factor, such
that
Hf ,...,Hf ∼A(f ) (5) 2.5. Threebasicpropertiesofselectiveinfluences
{a 1} {a n}
(cid:16) (cid:17)
foreverytreatmentf ∈T. For completeness, we list three other fundamental conse-
quencesofJDC(Dzhafarov&Kujala,2010).
SeeDzhafarovandKujala(2010)foraproof.ThevectorHin
theformulationoftheJDC isreferredtoastheJDC-vectorfor
A(f ), or the hypotheticalJDC-vector for A(f ), if the existence
2.5.1. Nestedness.
ofsuchavectorofjointlydistributedvariablesisinquestion.
The JDC promptsa simple justification for our definition of
selectiveinfluences.Let,forexample,(A,B,C)"(a ,b ,g ),with For any subset {i1,...,ik} of {1,...,n}, if (A1,...,An) "
a ={1a ,2a },b = 1b ,2b ,3b ,g ={1g ,2g ,3g ,4g }. Considerall (a 1,...,a n)then(Ai1,...,Aik)"(a i1,...,a ik).
treatmentsf inwhichthefactorpointofa isfixed,say,at1a . If
(A,B,C)"(a ,b ,g (cid:8)),theninth(cid:9)evectorsofrandomvariables
2.5.2. CompleteMarginalSelectivity
a b g a b g a b g
(A,B,C) 1 ,2 ,1 ,(A,B,C) 1 ,2 ,3 ,(A,B,C) 1 ,3 ,1
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) For any subset {i1,...,ik} of {1,...,n}, if (A1,...,An) "
themarginaldistributionofthevariableAisoneandthesame, (a ,...,a )thenthek-marginaldistribution4of(A ,...,A )(f )
1 n i1 ik
does not depend on points of the factors outside (a ,...,a ).
A 1a ,2b ,1g ∼A 1a ,2b ,3g ∼A 1a ,3b ,1g . Inparticular,thedistributionofA onlydependsonpio1intsofiak ,
i i
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) i=1,...,n.
But the intuition of selective influencesrequiresmore: thatwe
This is, of course, a trivial consequence of the nestedness
a
can denote this variable A(1 ) because it preserves its identity
property, but its importance lies in that it provides the easiest
(and not just its distribution) no matter what other variables it
tochecknecessaryconditionforselectiveinfluences.
b g b g b g
is pairedwith, (B,C) 2 ,1 , (B,C) 2 ,3 , or(B,C) 3 ,1 .
a b b g
Analogous statements hold for A(2 ), B 2 , B 3 , C(1 ),
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
etc. The JDC formalizesthe intuitive notion of variables“pre-
2.5.3. Invarianceunderfactor-point-specifictransformations
(cid:0) (cid:1) (cid:0) (cid:1)
serving their identity” when entering in various combinations
witheachother:therearejointlydistributedrandomvariables
Let(A ,...,A )"(a ,...,a )and
1 n 1 n
H1a ,H2a ,H1b ,H2b ,H3b ,H1g ,H2g ,H3g ,H4g
wishcoosmebidineendtitwyitihs draenfidnoemd bvyartihaibslejsoiHnt2bdiasntrdibHut1igo,ni;twfohremnsHth1ae H =(cid:18)Hxa11,...,Hxak1i,...,Hxa1n,...,Hxaknn(cid:19)
triad (H1a ,H2b ,H1g ) whose distribution is the same as that be the JDC-vector for (A1,...,An)(f ). Let F be any function
a b g
of (A,B,C) 1 ,2 ,1 ; when the same random variable H1a thatappliestoH componentwiseandproducesacorresponding
is combined(cid:0) with ran(cid:1)dom variables H2b aandb Hg3g , the triad vectorofrandomvariables
(H1a ,H2b ,H3g )isdistributedas(A,B,C) 1 ,2 ,3 ; andsoon
—thekeyconceptbeingthatitisoneandthesameH1a which F xa 1,H a ,...,F xa i,H a ,
idsobmeivnagripaabilreesdAwit1ha o,2thb e,r1gva,riAab1leas,,2abs,o3pg(cid:0)p,oAsed1ato,3d(cid:1)bif,f1egrenwthraicnh- F(H)= (cid:16) 1 x11(cid:17) ..., (cid:18) k1 xk1i(cid:19) ,
aardeeimdeonntsictraaltliyodnits(cid:0)htraitbtuhteedid.eS(cid:1)neteitDy(cid:0)zishnafoatrgoevn(cid:1)aenrda(cid:0)lKlyupjarelase(r2v(cid:1)0e1d0i)ffaolrl  F xa1n,Hxa1n ,...,F xaknn,Hxaknn 
 (cid:16) (cid:17) (cid:16) (cid:17) 
weknowismarginalselectivity(asdefinedinSection2.5).
ThefollowingisanimportantconsequenceofJDC.
Theorem2.3. InDefinition2.1,therandomentityRcanalways 4k-marginal distribution isthedistribution ofasubsetofkrandomvariables
(k≥1)inasetofn≥kvariables. InTownsendandSchweickert(1989)the
bechosentobearandomvariable. Moreover,Rcanbechosen
propertywasformulatedfor1-marginalsofapairofrandomvariables. The
arbitrarily,asanycontinuously(atomlessly)distributedrandom
adjective“complete”weusewith“marginalselectivity”istoemphasizethat
variable,e.g.,uniformlydistributedbetween0and1. wedealwithallpossiblemarginalsratherthanwithjust1-marginals.
SelectivityinProbabilisitcCausality 5
a
wherewedenotebyF(x ,·)theapplicationofF tothecompo- Until very recently (see footnote 1) we were blissfully un-
a
nentlabeledbyx . Clearly,F(H)possessesajointdistribution awareoftheanalogousdevelopmentsinquantumphysics. The
andcontainsonecomponentforeachfactorpoint.Ifwenowde- mostconspicuousparallelscanbefoundinFine(1981a-b),but
fineavectorofrandomvariablesB(f )foreverytreatmentf ∈T thatworkinturnbuildsonavenerablelineofresearchandthink-
as ing: goingbackfirsttoBell(1964),andultimatelyto Einstein,
Podolsky, and Rosen’s (1935)paper. The issue in question re-
(B1,...,Bn)(f )= F f {a 1},A1 ,...,F f {a n},An (f ), gardstwo“noncommuting”measurements,suchasthoseofthe
momentum and of the location of a particle, or spin measure-
then it follows from J(cid:0)DC(cid:0)that (B ,(cid:1)...,B )(cid:0)"(a ,...(cid:1),(cid:1)a ).5 A ments along two different axes. For our purposes it is suffi-
1 n 1 n
function F(xa i,·) can be referred to as a factor-point-specific cienttostatethatwhenoneoftwononcommutingmeasurements
transformation of the random variable A, because the random is performed(withoutuncertaintyaboutthe result), the second
i
variableistransformeddifferentlyfordifferentpointsofthefac- one cannotbe performedon the same system. The key insight
tor assumed to selectively influence it. We can formulate the neededtounderstandtheanalogywiththeproblemofselective
propertyinquestionbysayingthatadiagramofselectiveinflu- influencesisthis:noncommutingmeasurementsonthesamesys-
encesisinvariantunderallfactor-point-specifictransformations tem,beingmutuallyexclusive,canbeviewedaslevels(mutually
oftherandomvariables.Notethatthisincludesasaspecialcase exclusivevalues)ofoneandthesameexternalfactor.
transformationswhicharenotfactor-point-specific,with Thisisnotentirelyintuitive. Considertwoparticlesforeach
of which one can measure its momentum or its location. The
F xa i,· ≡...≡F xa i,· ≡F(a ,·). analogy requires that one view the measurement on particle 1
1 ki i asafactora 1 withtwomutuallyexclusivelevels,1a 1 (location
(cid:0) (cid:1) (cid:16) (cid:17) measurement)and2a 1 (momentummeasurement);andthemea-
This propertyis importantfor constructionand use of tests for
surementonparticle2isafactora withtwomutuallyexclusive
selectiveinfluences(Dzhafarov&Kujala,2010;Kujala&Dzha- 2
a a
levels,1 2 and2 2, interpretedanalogously. Thetwomeasure-
farov,2008).
a a a a
mentscanbecombinedintreatments,(1 1,1 2),(1 1,2 2),etc.,
a a a a
but not within a factor, (1 1,2 1) or (1 2,2 2). The results of
each of the measurementsis a randomvariable, A for particle
2.6. Quantumentanglementandselectiveinfluences 1
1andA forparticle2. ThepossiblevaluesA forA arepos-
2 1 1
sible locations of particle 1 if a 1 is at level 1a 1, but they are
In psychology, the notion of selective influences was intro- possible momentum values for particle 1 if a 1 is at level 2a 1
duced by Sternberg(1969),in the contextof studying“stages” (whichmakesitawkwardbutstillpossibletomaintainthecon-
of informationprocessing. Sternbergacknowledgedthatselec- ventionmentionedinfootnote2). Itiseasierwithspins(Bohm
tiveinfluencescanholdevenifthedurationsofthestagesbeing &Aharonov,1957): forinstance,forspin-1/2particles(suchas
selectively affected are not stochastically independent, but he electrons),A consistsoftwopossiblevaluesofspininonedi-
1
lacked the mathematical apparatus for dealing with this possi- rectionifa 1 isatlevel1a 1 andoftwopossiblevaluesofspinin
bility. Townsend(1984)wasthe firstto studythenotionofse- anotherdirectionifthelevelis2a 1. Thesetwotwo-elementsets
lectivenessunderstochasticinterdependencesystematically. He aremorenaturaltoconsider“thesame.”
proposed to formalize the notion of selectively influenced and
With all this in mind, the question now can be posed in the
stochastically interdependent random variables by the concept familiar to us form: can we say that (A ,A ) " (a ,a ), or
1 2 1 2
of “indirect nonselectiveness”: the conditional distribution of can the measurement (factor) a influence the result (random
1
the variable A1 given any value a2 of the variable A2, depends variable)A and/ora influenceA ? IntheEinstein-Podolsky-
ona only,and,bysymmetry,theconditionaldistributionofA 2 2 1
1 2 Rosen (EPR) paradigm involving entangled particles, the two
at any A =a depends on a only. Under the name of “con-
1 1 2 randomoutcomesA ,A are stochastically interdependent,and
1 2
ditionally selective influence” this notion was mathematically
theirjointdistributionateverytreatmentis(correctly)predicted
characterized and generalized in Dzhafarov (1999). It turned
bythequantumtheory.Thequestionthereforebecomes:arethe
out, however,that this notion couldnot serve as a generaldef-
predicted(andobserved)jointdistributionsof(A ,A )compati-
1 2
inition of selective influences, because it did not satisfy some blewiththehypothesis(A ,A )"(a ,a )? Einstein,Podolsky,
1 2 1 2
intuitive desiderata for such a definition, e.g., the nestedness andRosen(1935)took(A ,A )"(a ,a )forgrantedifthetwo
1 2 1 2
and marginal selectivity properties formulated in Section 2.5.
particlesareseparatedinspaceandmeasuredsimultaneously(in
VariantsofDefinition2.1ofthepresentpaperwereproposedin
someinertialframeofreference).
Dzhafarov(2003)andbothelaboratedandgeneralizedinDzha-
Bell’s (1964) celebrated theorem shows that (A ,A ) "
farovandGluhovsky(2006),KujalaandDzhafarov(2008);JDC 1 2
was explicitly formulated in Dzhafarov and Kujala (2010), al- (a 1,a 2)isnotthecase forentangledspin-1/2 particlesobeying
the laws of quantummechanics. The reason this result is con-
thoughclearlyimpliedintheearlierwork.
sidered to be of foundational importance (“the most profound
discoveryin science,” repeatingthe oft-quotedcharacterization
byStapp,1975)isthatBellessentiallyadoptedDefinition2.1for
(A ,A )"(a ,a )andidentifiedtherandomentityRwiththe
5SinceitispossiblethatF(xa ,Hxa )andF ya ,Hya ,withxa 6=ya ,havedif- set1of2allhidde1nv2ariablesofaconceivabletheory“explaining”
ferentsetsofpossiblevalues,strictlyspeaking,onemayneedtoredefinethe
functionstoensurethatthesetsofpossibl(cid:0)evalues(cid:1)forB(f )isthesamefor the dependence of (A1,A2) on (a 1,a 2): knowing a value of R
differentf .Thisis,however,notessential(seefootnote2). onewouldbeabletopredict,throughthefunctions f1 and f2 of
6 DzhafarovandKujala
Definition2.1,thevaluesof(A ,A ). Inadditiontobeingcalled geted when (A ,A )"(a ,a ) is rejected, and distinguishing
1 2 1 2 1 2
“hidden”thevariablesentailedinRarereferredtoas“context- thetwoisachallenge.
independent”(meaningthat the distribution of R and the func- FollowingthelogicofBell’swork,Clauser,Horne,Shimony,
tions f ,f donotdependontreatments)and“local”(meaning, & Holt (1969) derived a system of inequalities that are neces-
1 2
essentially,thatinthetheoryinvolvingRand f ,f themeasure- sary conditions for (A ,A ) "(a ,a ) in the EPR paradigm
1 2 1 2 1 2
ment a does not influence A , nor a influences A ). Bell’s with two particles and two measurements(factors) with binary
1 2 2 1
(1964)theoremthereforeis interpretedas stating that quantum outcomes. TheseinequalitiesaresubsumedinFine’s(1982a-b)
predictionsregardingtwoentangledspin-1/2particlescannotbe ones (discussed in Section 3.5), which present both necessary
explainedbyanytheoryinvolvingcontext-independentandlocal andsufficientconditionsfor(A ,A )"(a ,a ),basedonJDC.
1 2 1 2
variables. ThelatterwasintroducedinFine’spapersforthefirsttime(and
A rejection of (A ,A )"(a ,a ) in quantum physics can called by this name too), althoughthe earlier Suppesand Zan-
1 2 1 2
be handledbydispensingwith locality(Bohm’sapproach),but otti’s (1981)Theoremon Common Causes can also be viewed
mostphysicistsfindthisuntenable(measurementa cannotin- asaspecialformofJDC.
1
fluence A if they are separated by a space-like interval). The Fine’sinequalitiesformaspecialcaseoftheLinearFeasibility
2
quantumprobabilitytheorycanbeviewedasawayofallowing Test considered in the nextsection. We therefore defer further
for context-dependencewhile retaining locality. In behavioral discussionoftheEPRparadigmtoSection3.5,andconcludethe
applicationsbothlocalityandcontext-independencecanbetar- presentsectionbythefollowingtableofcorrespondences:
SelectiveProbabilisticCausality QuantumEntanglementProblem(forspins)
observedrandomoutput detectedspinvalueofagivenparticle
factor/input spinmeasurementinagivenparticle
factorlevel setting(axis)ofthespinmeasurement
jointdistributioncriterion jointdistributioncriterion
canonicaldiagramofselectiveinfluences “classical”explanation(bycontext-independentlocalvariables)
3. LINEARFEASIBILITYTEST weknowthatthesetofpossiblevaluesfortherandomvariable
H is ai,...,ai ,irrespectiveof j. Denote
xij 1 mi
InthissectionweassumethatforeachrandomvariableA(f )
i (cid:8) (cid:9)
einm(eAnt1s,,..a.i,,A.n.).(,fai) .theItsiestaArgiuoafbiltystphoessmibolset vimalpuoerstahnatssmpeiceial-l Pr A1=a1l1,...,An=anln x1j1,...,xnjn
1 mi h(cid:16) (cid:17)(cid:16) (cid:17)i
casebothbecauseitisubiquitousandbecauseinallothercases
forr.v.s forfactorpoints (6)
randomvariablescan be discretizedinto finite numberof cate-
gories. We are interested in establishing the truth or falsity of =P l1,...,ln; j1,...,jn ,
 
thediagramofselectiveinfluences(3),whereeachfactora in
j z }| { z }| {
(a ,...,a ) contains k factor points xj,...,xj (written so in-  
1 n aj a 1 kj whereli∈{1,...,mi}and ji∈{1,...,ki}fori=1,...,n(“r.v.s”
stead of more formal x j,...,x j). The Linear Feasibility Test abbreviates“randomvariables”).Denote
1 kj
(LFT)to bedescribedisa directapplicationofJDC tothissit-
H =a1 ,...,H =a1 ,
udaiatgiornam, fuorfnsieslheicntgiveainneflcueesnsacreys(aAn1d,.s.u.f,fiAcni)en"t c(oa n1d,i.t.io.,na fno)r. the Pr x11 l11 ..., x1k1 l1k1 
 Hxn1 =anln1,...,Hxnkn =anlnkn 
(7)
 
3.1. Thetest
forA1 forAn
=Q l ,...,l ,...,l ,...,l ,
InthehypotheticalJDC-vector 11 1k1 n1 nkn
z }| { z }| {
 
 
H=(cid:18)Hx11,...,Hx1k1,...,Hxn1,...,Hxnkn(cid:19), mwkhnerQe-lpijro∈b{a1b,il.i.ti.e,sm.i}Aforerqiu=ire1d,.j.o.i,nnt.dTishtirsibguivtieosnufsomrk1t1he×J.D..C×-
n
sinceweassumethat,foranypointxi offactora iandanytreat- vector H exists if and only if these probabilities can be found
mentf containingxi, j subjecttomk11×...×mknn nonnegativityconstraints
j
Hxij ∼Ai(f ), Q(l11,...,l1k1,...,ln1,...,lnkn)≥0, (8)
SelectivityinProbabilisitcCausality 7
and(denotingbyn thenumberoftreatmentsinT)n ×m × Theorem 3.1. The rank of M for a maximal set of treatments
T T 1
...×m linearequations T =a ×...×a is
n 1 n
(cid:229) Q(l11,...,l1k1,...,ln1,...,lnkn) (k1(m1−1)+1)...(kn(mn−1)+1).
(9)
=P(l ,...,l ;j ,...,j ), Proof. Givenany
1 n 1 n
where the summation is across all possible values of the {1′,...,s′}⊂{1,...,n},
(l ,...,l ,...,l ,...,l )subjectto
11 1k1 n1 nkn (j1′,...,js′)∈{1,...,k1′}×...×{1,...,ks′},
l1j1 =l1,...,lnjn =ln.6 (l1′,...,ls′)∈{1,...,m1′}×...×{1,...,ms′},
This can be more compactly formulated in a matrix form. let V(1′,...,s′;j1′,...,js′;l1′,...,ls′) denote an
Let the observable probabilities P(l ,...,l ;j ,...,j ) consti-
1 n 1 n (m )k1...(m )kn-component Boolean row vector whose
tute components of a n ×m ×...×m -dimensional col- 1 n
T 1 n components are lexicographically enumerated in the same
umn vector P, with its cells lexicographically enumerated
way as Q, and such that its (l ,...,l ,...,l ,...,l )th
by (l1,...,ln;j1,...,jn). Let the hypothetical probabili- componentis1ifandonlyif 11 1k1 n1 nkn
ties Q(l ,...,l ,...,l ,...,l ) constitute components of a
11 1k1 n1 nkn
imcok11g×ra.p.h.i×camllyknne-dniummeenrsaitoendablyco(llum,.n.v.,elctor,.Q..,,wl it,h..it.s,lcell)s.lLexe-t l1′j1′ =l1′,...,ls′js′ =ls′.
11 1k1 n1 nkn
MbeaBooleanmatrixwithnT×m1×...×mnrowsandmk11× The rows of matrix M are V(1,...,n;j1,...,jn;l1,...,ln)-
...×mkncolumnslexicographicallyenumeratedinthesameway vectors. It is easy to check that for any fixed
n
as, respectively, P and Q, such that the entry in the cell in the (1′,...,s′;j1′,...,js′), the sum ofthe rowsofM corresponding
(l1,...,ln;j1,...,jn)th row and (l11,...,l1k1,...,ln1,...,lnkn)th tofixedvalues(l1′,...,ls′)isV(1′,...,s′;j1′,...,js′;l1′,...,ls′).
column is 1 if l1j1 =l1,...,lnjn =ln; otherwise the entry is 0. It follows that for s = n,n − 1,...,1, a vector
Clearly,thevectorQexistsifandonlyifthesystem V(1′,...,s′;j1′,...,js′;l1′,...,ls′) in which all li′ = 1 ex-
ceptfori′∈{1′′,...,v′′}⊂{1′,...,s′}(asubsetofv<sdistinct
MQ=P, Q≥0 (10)
elements),isalinearcombinationofthevector
(withtheinequalityunderstoodcomponentwise)hasasolution.
Thisis a typicallinearprogramming(LP)problem. More pre- V 1′′,...,v′′;j1′′,...,jv′′;l1′′,...,lv′′
cisely,thisisanLPtaskinthestandardformandwithadummy
(cid:0) (cid:1)
objective function (e.g., a linear combination with zero coeffi- andallthevectors
cients). Itisknown(Karmarkar,1984;Khachiyan,1979)thatit
isalwayspossible,inpolynomialtime, toeitherfinda solution V 1′,...,s′;j1′,...,js′;l1′,...,ls′
for such a system or to determine that it does not exist. Many
(cid:0) (cid:1)
standardsoftwarepackagescanhandlethisproblem(e.g.,GNU forwhichallli′ >1and
LinearProgrammingKitathttp://www.gnu.org/software/glpk/).
{j1′′,...,jv′′;l1′′,...,lv′′}⊂{j1′,...,js′;l1′,...,ls′}.
3.2. PropertiesoftheLPproblem AsaresulttherowsofMarelinearcombinationsoftherowsof
M∗ consistingofvectors
TherankofmatrixMisalwaysstrictlysmallerthanthenum-
berofcomponentsinP. Thisfollowsfromthefactthatforany V 1′,...,s′;j1′,...,js′;l1′,...,ls′
two allowabletreatments(j1,...,jn) and(j1′,...,jn′) thatshare forallpossible (cid:0) (cid:1)
asubvector
(j1′,...,js′)= j1′′,...,js′′ {1′,...,s′}⊂{1,...,n},
(whereweuse{1′,...,s′}todesig(cid:0)natesdistin(cid:1)ctelementschosen (j1′,...,js′)∈{1,...,k1′}×...×{1,...,ks′},
from {1,...,n}), and for any fixed (v1,...,vs), the sum of all (l1′,...,ls′)∈{2,...,m1′}×...×{2,...,ms′}.
rowsofMcorrespondingto(l ,...,l ;j ,...,j )thcomponents
1 n 1 n
ofPwith(l1′,...,ls′)=(v1,...,vs)isthesameBooleanvectoras Bystraightforwardcombinatoricsthenumberofsuchvectorsis
thesumofallrowsofMcorrespondingto(l ,...,l ;j′,...,j′)th
1 n 1 n
componentsofPwiththesameproperty.Theupperlimitforthe (k1(m1−1)+1)...(kn(mn−1)+1).
rankofmatrixMisgiveninthefollowingtheorem.
The rows of M∗ are linearly independent
because the column corresponding to the
(l =1,...,l =1,...,l =1,...,l =1)th component
11 1k1 n1 nkn
6ThesumofallQ’sis1becauseitequalsthesumofallP’s(acrossalll1,...,ln) of Q contains a single 1, in the row of M∗ corresponding to
foranygiventreatment j1,...,jn. s=0(whichrowcontains1’sonly).
8 DzhafarovandKujala
Notethat
Marginalselectivityhereissatisfiedtrivially:allmarginalprob-
ki(mi−1)+1<mkii abilitiesareequal0.5, foralltreatments. Inthe matrixformof
theLFT,thecolumn-vectoroftheabove16probabilities,
forallk ≥2andm ≥1. Thismeansthat
i i
(k (m −1)+1)...(k (m −1)+1)<(m )k1...(m )kn, (.140,.360,.360,...,.040,.040,.460)⊤,
1 1 n n 1 n
using ⊤ for transposition, is denoted by P. The LFT problem
andthesystemMQ=Pisalwaysunderdetermined.
is defined by the system MQ=P, Q≥0, where the 16×16
Corollary 3.2. If P satisfies marginal selectivity, then system Boolean matrix M is shown below: each column of the ma-
(10)isequivalentto trix corresponds to a combination of values for the hypotheti-
calH-variables(shownabovethematrix),whileeachrowcorre-
M∗Q=P∗, Q≥0, (11) spondstoacombinationofatreatmentwithvaluesoftheoutputs
A,B(shownontheleft).
where M∗ is as defined in the proof above, and P∗ is the “re-
ducedhierarchical”vectorwithcomponents
Pr A1′ =a1l1′′,...,As′ =asls′′ x1j1′′,...,xsj′s′ (12) HH12aa 11 11 11 11 12 12 12 12 21 21 21 21 22 22 22 22
h(cid:16) =P1∗′,...,s′(l1′,..(cid:17).,(cid:16)ls′;j1′,...,js(cid:17)′)i, H1b 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
wheres=0,...,n,{1′,...,s′}⊂{1,...,n},andli′ ∈{2,...,mi} H2b 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
foreachi′∈{1′,...,s′}. M∗ isoffullrowrank. a b A B
1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0
To comment on this corollary, it follows from the proof of
1 2 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0
Theorem3.1thatMQ=PneverhasasolutionifvectorPvio-
latestheequality 2 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0
2 2 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1
(cid:229) Pr A1=a1l1,...,An=anln x1j1,...,xnjn 1 2 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0
=(cid:229) hP(cid:16)r A =a1,...,A =a(cid:17)n(cid:16) x1 ,...,x(cid:17)ni , 1 2 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0
1 l1 n ln j1′ jn′
h(cid:16) (cid:17)(cid:16) (cid:17)i 2 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0
where the summation is across all values of (l ,...,l ) with a
1 n 2 2 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
fixed (l1′,...,ls′). Clearly, this necessary condition is just an- 2 1 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0
otherwayofstatingmarginalselectivity. AssumingthatPdoes
satisfymarginalselectivity,itcanberepresentedbythe“reduced 1 2 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0
hierarchical”vectorP∗ whosecomponentsare marginalproba- 2 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0
bilitiesofallorders,withs=0correspondingtotheprobability 2 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1
1.
2 2 1 1 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0
1 2 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0
3.3. Examples 2 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0
2 2 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1
Example 3.3. Let a ={1a ,2a }, b = 1b ,2b , and the set of
allowabletreatmentsT consistofallfourpossiblecombinations The linear programing routine of MathematicaTM(using the
of the factor points. Let A and B be B(cid:8)ernoull(cid:9)i variables, a = interiorpointalgorithm)showsthatthelinearequations(9)have
1
b =1,a =b =2,distributedasshown: nonnegativesolutionscorrespondingtotheJDC-vector
1 2 2
a b A B Pr a b A B Pr H1a H2a H1b H2b Pr H1a H2a H1b H2b Pr
1 1 1 1 .02708610 2 1 1 1 .15748000
1 1 1 1 .140 1 2 1 1 .198
1 1 1 2 .00239295 2 1 1 2 .00204128
1 2 .360 1 2 .302
1 1 2 1 .16689300 2 1 2 1 .10854100
2 1 .360 2 1 .302
1 1 2 2 .03358610 2 1 2 2 .00197965
2 2 .140 2 2 .198
1 2 1 1 .00197965 2 2 1 1 .03358610
a b A B Pr a b A B Pr 1 2 1 2 .10854100 2 2 1 2 .16689300
2 1 1 1 .189 2 2 1 1 .460 1 2 2 1 .00204128 2 2 2 1 .00239295
1 2 .311 1 2 .040 1 2 2 2 .15748000 2 2 2 2 .02708610
2 1 .311 2 1 .040 Thecolumn-vectoroftheseprobabilitiesconstitutesQ>0.This
2 2 .189 2 2 .460 provesthatinthiscasewedohave(A,B)"(a ,b ).
SelectivityinProbabilisitcCausality 9
Example3.4. Inthepreviousexample,letuschangethedistri- P(l ,l ,...,l ;j ,j ,...,j )areredefinedas7
1 2 n 1 2 n
butionsof(A,B)tothefollowing:
P′(l ,l ,...,l ;j ,j ,...,j )
1 2 n 1 2 n
1P(l ,l ,...,l ;1,j ,...,j )+1P(l ,l ,...,l ;2,j ,...,j )
2 1 2 n 2 n 2 1 2 n 2 n
if j =1,
a b A B Pr a b A B Pr = 1
P(l1,l2,...,ln;j1+1,j2,...,jn)
1 1 1 1 .450 1 2 1 1 .105 if j >1.
1
1 2 .050 1 2 .395 When we average the original equations for
2 1 .050 2 1 .395
P(l ,l ,...,l ;1,j ,...,j ) and P(l ,l ,...,l ;2,j ,...,j ),
1 2 n 2 n 1 2 n 2 n
2 2 .450 2 2 .105
weget
a b A B Pr a b A B Pr
1(cid:229) Q(l =l ,l ,...,l ,...,l ,...,l )
(cid:229) 2 l12 11 1 12 1k1 n1 nkn
2 1 1 1 .170 2 2 1 1 .110 (+21(cid:229) l11Q(l11,l12=l1...,l1k1,...,ln1,...,lnkn))
1 2 .330 1 2 .390
=P′(l ,l ,...,l ;1,j ,...,j ),
1 2 n 2 n
2 1 .330 2 1 .390
2 2 .170 2 2 .110 wherel2j2 =l2,...,lnjn =ln andthe outersummationis across
alll exceptforthefollowingvaluesfor(i,j):(1,1),(1,2),and
ij
(i,j),i=2,...,n.WedefineanewvectorQ′whosedimension-
i
alityislessthanthatofQbyone,putting
Once again, marginal selectivity is satisfied trivially, as all
marginal probabilities are 0.5, for all treatments. The linear
Q′(l =l,l ,...,l ,...,l ,...,l )
programingroutineofMathematicaTM,however,showsthatthe 11 13 1k1 n1 nkn
= 1(cid:229) Q(l =l,l ,l ,...,l ,...,l ,...,l )
linear equations (9) have no nonnegative solutions. This ex- 2 l12 11 12 13 1k1 n1 nkn
cludes the existence of a JDC-vector for this situations, ruling +12(cid:229) l11Q(l11,l12=l,l13,...,l1k1,...,ln1,...,lnkn),
outtherebythepossibilityof(A,B)"(a ,b ).
where l has the same range as any of l . (For notationalsim-
1j
plicity, in Q′ we do not re-enumerate (1,3) as (1,2), (1,4) as
(1,3),etc.,leavingtherebyl undefined)
12
For any point of factor a other than x1 and x1, say, x1, we
3.4. Renamingandgrouping 1 1 2 3
havethen
SinceLFTisbothanecessaryandsufficientconditionforse- (cid:229) l11,l12Q(l11,l12,...,l1k1,...,ln1,...,lnkn)
lective influences, if it is passed for (A1,...,An)(f ), it is guar- =P(l1,l2...,ln;3,j2...,jn),
anteedtobepassedfollowinganyfactor-point-specifictransfor-
mations of these random outputs. All such transformations in whichcanbepresentedas
thecase ofdiscreterandomvariablescanbedescribedascom-
binationsofrenaming(factor-pointspecificone)andcoarsening 1(cid:229) Q(l =l,l ,l =l ,...,l ,...,l ,...,l )
(cid:229)(cid:229) 2 l12 11 12 13 1 1k1 n1 nkn
(sgimropulpyindgoeosfnsoomt deevpaelnudesotnogtehtehevra)l.uIensfoafctt,hteheraonudtcoommveaorfiaLbFleTs l( +12(cid:229) l11Q(l11,l12=l,l13=l1,...,l1k1,...,ln1,...,lnkn))
=P(l ,l ...,l ;3,j ...,j ).
involved,onlytheir probabilitiesmatter. Thereforea renaming 1 2 n 2 n
will not change anythingin the system of linear equationsand
Thisisequivalentto
inequalities(8)-(9). An exampleof coarseningwill be redefin-
ing A and B, each having possible values 1,2,3,4, into binary
variables (cid:229) Q′(l ,l =l ,...,l ,...,l ,...,l )
11 13 1 1k1 n1 nkn
=P′(l ,l ...,l ;j =2,j ...,j ),
1 2 n 1 2 n
1 ifA(f )=1,2, 1 ifB(f )=1,2,3,
A∗(f )= B∗(f )= wherel =l ,...,l =l ,andthesummationisacrossalll
(2 ifA(f )=3,4, (2 ifB(f )=4. 2j2 2 njn n ij
exceptfor (i,j)=(1,3) and (i,j)=(i,j), i=2,...,n. So we
i
haveobtainedasolutionforthefactor-coarsenedsystemfroma
Itisclearthatanysucharedefinitionamountstoreplacingsome solutionfortheoriginalsystem.
oftheequationsin(9)withtheirsums. Therefore,iftheoriginal
systemhasasolution,sowillalsothesystemaftersuchreplace-
ments. Of course, thereverseisnotgenerallytrue: the coarser
systemcanhavesolutionswhentheoriginalsystemdoesnot. 7More general mixtures, p P(l1,l2,...,ln;1,j2,...,jn) +
The same is true for coarsening the system by grouping to- (1−p )P(l1,l2,...,ln;2,j2,...,jn) for 0<p ≤1, are dealt with as eas-
ily; moreover, p =1formally corresponds todropping thefactor point x1,
gether some of the factor points within factors. Suppose we 2
want to group together points x1 and x1 of factor a contain- consideredbelow. Thevaluesofp otherthan1/2and1canbeusefulifthe
1 2 1 groupingisdoneonasamplelevel,toreflectthedifferencesinsamplesizes
ing more than two points. This means that the probabilities correspondingtotreatmentscontainingx1andx1.
1 2
10 DzhafarovandKujala
Droppingapoint,say,x1 isevensimpler: wedeleteallrows axes per particle k =k =2, and the numberof possible spin
2 1 2
with j =2,andthenredefinetheQvectoras valuesperparticle m =m =2 (thisvalue beingthe same for
1 1 2
allspin axeschosenfora givenparticle). Theycanbe written,
Q′(l11,l13,...,l1k1,...,ln1,...,lnkn) withreferenceto(6)and(12),as
=(cid:229) Q(l ,l ,l ,...,l ,...,l ,...,l ).
l12 11 12 13 1k1 n1 nkn −1≤P(2,2; j ,j )+P 2,2; j′,j
1 2 1 2
Iftherandomvariablesinvolvedhavemorethanfinitenumber
ofvaluesand/orthefactorsconsistofmorethanfinitenumberof +P 2,2; j1′,j2′ −(cid:0)P 2,2; j1,(cid:1)j2′
factorpoints,orifthesenumbers,thoughfinite,aretoolargeto −P1∗(cid:0) 2; j1′ −P(cid:1)2∗(2;(cid:0)j2)≤0, (cid:1)
handletheensuinglinearprogrammingproblem,thenLFTcan
stillbeusedafterthevaluesoftherandomvariablesand/orfac- where j1,j1′ ∈{1,2}, j2(cid:0),j2′ ∈(cid:1){1,2}, j16= j1′, j26= j2′. Thesein-
torshavebeenappropriatelygrouped.LFTthenbecomesonlya equalities constitute the necessary and sufficientconditionsfor
necessaryconditionforselectiveinfluences(withrespecttothe (A1,A2)"(a 1,a 2), with marginalselectivity assumed implic-
originalsystemoffactorsandrandomvariables),anditsresults itly.AlthoughFine’sderivationoftheseinequalitiesisdifferent,
willgenerallybedifferentfordifferent(non-nested)groupings. theycan be derivedas solutionsof system (11), with P∗ the 9-
componentvector(using⊤fortransposition)
Example3.5. Considerthehypothesis(A,B)"(a ,b )withthe
factorshavingafinitenumberoffactorpointseach,andAandB (1,P∗(2;1),...,P∗(2;2),P(2,2;1,1),...,P(2,2;2,2))⊤,
beingresponsetimes.TouseLFT,onecantransformtherandom 1 2
variableAas,say,
Qthe16-componentvector
1 ifA(f )≤a1/4(f ), (Q(1,1,1,1),...,Q(2,2,2,2))⊤,
2 ifa (f )<A(f )≤a (f ),
A∗(f )= 1/4 1/2
3 ifa1/2(f )<A(f )≤a3/4(f ), andM∗ thecorresponding9×16Booleanmatrix,
4 ifA(f )>a (f ),
3/4
andtransformBas H1a 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
1 ifB(f )≤b (f ), H2a 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
B∗(f )=(2 ifB(f )>b11//22(f ), HH12bb 11 12 21 22 11 12 21 22 11 12 21 22 11 12 21 22
where a (f ) and b (f ) designate the pth quantiles of, respec- a b A B
p p
tively A(f ) and B(f ). The initial hypothesis now is reformu- · · · · 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
lated as (A∗,B∗)"(a ,b ), with the understanding that if it is
1 · 2 · 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
rejectedthentheinitialhypothesiswillberejectedtoo(aneces-
2 · 2 · 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
sary conditiononly). LFT will now be applied to distributions
oftheform · 1 · 2 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
· 2 · 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 2 2 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1
a b A B Pr
1 2 2 2 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
x y 1 1 p
11 2 1 2 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1
1 2 p
12 2 2 2 2 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1
. . .
. . .
. . .
In fact, using a standard facet enumeration program (e.g.,
4 1 p
41
lrsprogramathttp://cgm.cs.mcgill.ca/~avis/C/lrs.html)thesein-
4 2 p
42 equalities(togetherwiththeequalitiesrepresentingmarginalse-
lectivity) can be derived “mechanically.” The essence of the
where the marginals for A are constrained to 0.25 and the
computation is in the fact that a linear system (10) or (11) is
a b
marginals for B to 0.5, for all treatments x ,y , yielding a feasible if and only if the pointP (respectively, P∗) belongsto
trivial compliance with marginal selectivity. Note that the test
theconvexhullofthepointscorrespondingtothecolumnsofM
mayverywelluphold(A∗,B∗)"(a ,b )eve(cid:8)nifma(cid:9)rginalselec- (respectively,M∗),whichformasubsetoftheverticesofaunit
tivity is violated for (A,B)(f ) (e.g., if the quantiles ap xa ,yb hypercube. Thefacetenumerationprogramsderiveinequalities
b
changeasafunctionofy ). describingthisconvexhull.
(cid:0) (cid:1)
Given a set of numerical (experimentallyestimated or theo-
retical)probabilities,computingtheLPproblem(10)or(11)is
3.5. Quantumentanglement always preferable to dealing with explicit inequalities as their
number becomes very large even for moderate-size vectors P.
Fine’s (1982a-b) inequalities relate to the simplest EPR While Fine’s inequalitiesfor n=2, k =k =2, m =m =2
1 2 1 2
paradigm, with the number of particles n=2, number of spin (assumingmarginalselectivity)numberjust8,alreadyforn=2,