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The British Society for the Philosophy of Science
Black Box Inference: When Should Intervening Variables Be Postulated?
Author(s): Elliott Sober
Source: The British Journal for the Philosophy of Science, Vol. 49, No. 3 (Sep., 1998), pp. 469-
498
Published by: Oxford University Press on behalf of The British Society for the Philosophy of
Science
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Brit.J. Phil.Sci.49 (1998),469-498
Black
Box
Inference:
When
Should
Intervening
Variables
Be
Postulated?
Elliott Sober
AB STRACT
An empirical procedureis suggested for testing a model that postulates variables
that intervene between observed causes and observed effects against a model that
includes no such postulate. The procedure is applied to two experiments in
psychology. One involves a conditioning regimen that leads to response general-
ization; the other concerns the question of whether chimpanzees have a theory of
mind.
1 Introduction
2 Onecauseandoneeffect
3 Multiplecauses,
multipleeffects,andmultiple
intervening
variables
4 Multiplecauses,multipleeJ%ects,
anda singleintervening
variable
S Conditional
independence
6 Multiple
layersof intervening
variables
7 Background
assumptions
in blackboxinference
8 Leatning
9 Responsegeneralization
10
Chimpanzee
theoryof mind
Concluding
comments
1 Introduction
You observe a numberof causes impinge on a system. You also observe the
systemgeneratea numberof effects. Whenshouldyou inferthatthecauses are
linkeddirectlyto the effects, andwhen shouldyou inferthatcauses arelinked
to effects by passing throughinterveningvariables?
Perhapsthemostfamiliarinstanceof thisproblemis thedebatein psychology
between mentalism and radical behaviourism.Mentalists hold that beliefs,
desires, and sensations are caused by environmental contingencies
and
themselves cause behaviour. Radical behaviourists claim that there is no
need to postulate such interveningvariables that a model linking stimulus
(C)OxfordUniversityPress 1998
470
Elliott Sober
directlyto responseis preferable.1Howeversalientthis examplemightbe, the
problemof blackbox inferenceis moregeneral.The 'system' on which causes
impinge and that generateseffects may be an organism,but it also may be
bigger or smaller.Economiesexpenence inputsandoutputs;so do organsand
cells. The problemof black box inferenceisn't just aboutorganismsandtheir
behaviour;it also anses at higherand lower levels of organization.
Justas blackbox inferenceoccursin domainsoutsideof psychology, so it is
not restncted to the question of whether an interveningvariable should be
postulated.To be sure, it is naturalto begin with the question of choosing
betweenOsuchpostulatesand 1. However,it wouldnot be surpnsingif similar
issues arise when the question is 2 versus 1. The question of
whether
to
postulate an interveningvariable is a special case of the question of
how
many
such vanables a theory should invoke. This means thatthe problemof
black box inference should interestthe convinced cognitivist. Even if it is a
mistake to forswear intervening vanables unconditionally,it still is worth
asking when and why interveningvanables shouldbe introduced.
2 One cause and one effect
Letusbegin,then,withthefollowingsimpleproblem.Youobserveonecause(C)
andone effect (E),andyourquestionis whichof themodelsin Figure1 is better.
(N-1, 1)
C
sE
(IV-1, 1, 1)
C
>
I
* E
Fig 1
In describingradicalbehaviouristsas decliningto postulateinterveningvariables,I mainlyhave
in mind the position of B.F. Skinner [1938, 1950]. It should be noted, however, that many
behaviouristshad no problemwith the idea of interveningvariablesand that Skinnerhimself
occasionally said thathe had nothingagainstthem.
E. C. Tolmanintroducedthe tellll 'interveningvariable'into psychology.Manypsychologists
now have the impression,due largely to the influentialarticle by MacCorquodaleand Meehl
[1948], thatTolmanthoughtthatinterveningvariablesmustbe operationallydefinedin termsof
observables. However, Tolman [1948] explicitly rejected that interpretation;see Amundson
[1983] and Smith [1986] for furtherdiscussion. ClarkHull [1943] also held that intervening
variables are legitimate theoreticalentities, although he saw no reason to interpretthem as
cognitive representations.
The intervening variables postulated by the models I'll discuss in what follows are not
operationallydefined, in the sense that their existence and states cannot be
deduced
from
input and output data. Rather,the frameworkis
probabilistic;
interveningvariable models
make probabilisticpredictionsthat differ from those made by models that do not postulate
interveningvariables.This probabilisticapproachwas anythingbut alien to Tolman;see, for
example, Tolman and Brunswik [1935] and Brunswik [1943]. Brunswik was influencedby
Reichenbach(as am I); TolmanandBrunswikdiscoveredthattheirapproacheshad a greatdeal
in common (Smith [1986]).
BlackBoxInference:WhenShouldInterveningVariablesBe Postulated? 471
I call the firstmodel 'N- 1, 1' becauseit postulatesno interveningvariableand
describestherelationof one cause andone effect. The secondmodelpostulates
an interveningvariablethatconnects one cause to one effect.
Youmightbe temptedto optfortheinterveningvariablemodelby appealingto
the pnnciple of
no actionat a distance.
If the events C and E are temporally
separated,then the causalconnection
of C to E must be mediatedby a causal
process in which there are inteimediatelinks (Hull [1943]; Hempel [1965],
pp. 203-4). To this argument,it may be replied that the (N-1, 1) model is
committedto actionat a distanceno morethan(IV-1, 1, 1) is. The lattermodel
neglects to assertthatthe connectionof C to I and the connectionof I to E is
mediatedby furEerevents,butthatis not the sameas denyingthatsuchevents
exist.Thedistinctionbetweenagnosticismandaieism needstobebomeinmind.
AlthoughI thinkthatthis reply is coITect,2it has the consequencethat the
two models are not incompatible;rather,the relationshipis that (IV-1, 1, 1)
entails (N-1, 1), but not conversely. This raises the questionof whetherthey
are,properlyspeaking,competitors,a questionthatI will not pursuefurther.I
will note, however,thatif they do not competewith each other,thentherecan
be no reason(otherthanpragmaticreasonsof convenience) to acceptthe one
and rejectthe other.
One factor that does separatethe two models is parsimony the (N-1, 1)
model postulatesfewer events and causal connections than the (IV-1, 1, 1)
model. In conventional scientific practice, this fact about the two models is
taken to confer on the (N-1, 1) model the status of a
nullhypothesis
it is
presumedinnocentuntil proved guilty. If the two models fit the observations
about equally well, you should opt for (N-1, 1); you should abandonthe
simplermodelin favourof themorecomplexalternativeonly if the (IV-1, 1, 1)
model does a betterjob of accommodatingthe observations.
The question, then, is whether the intervening-variablemodel fits the
observationsbetter.Let's begin by characterizingthe observations.In black
box inference, we observe frequencies
f(C), f(not-C), f(E]C), f(E]not-C),
f(E), and f(not-E). From these, we must infer values for the probabilities
postulatedbythemodels.Theusualprocedure
likelihoodestimation.3
is theonethatmaximizestheprobability
of the observations.Forexample,if E occurs72%of thetime whenC occursin
your observationalsample, then the best estimateof Pr(EIC)is 0.72.
2
If an arrowin a causalmodel entailedthattheremustbe aninterveningvariable,this wouldhave
the consequencethatcausalchainsmustbe dense ratherthanquantized.I see no reason
to think
thatcausalmodelsin differentsciences automaticallyhave commitmentson this fairlyrecondite
matter.
3
Since maximumlikelihoodestimationignorespriorprobabilities,Bayesiansdo not thinkmuch
of it. However,the pointsI'll makein what follows still standwithina Bayesianframework,so
long as you distinguishwhatthe currentdatatell you aboutthe competingmodelsfromthe prior
informationyou have aboutthem.
is maJcimum
Thebestestimateof a setof probabilities
472
Elliott Sober
The thingto notice aboutmaximumlikelihoodestimationin the case of the
(N-1, 1) model is that there is a
unique
maximum-likelihoodassignmentof
values.Foreach observedfrequency,thereis a probabilityin the model whose
value needsto be fixed,andthereis one suchassignmentthatdoes thejob best,
namely the one in which the probabilityis assigned a value that matchesthe
observedfrequency.
The (IV-1, 1, 1)modelis different.Forexample,if E occurs72%of thetime
when C occurs in a set of observations,the maximumlikelihood estimate of
Pr(EgC)is 0.72. However, accordingto the intervening-vanablemodel, that
probabilityis a functionof threeotherindependentquantities:
Pr(EIC) = Pr(EII)Pr(IIC) + Pr(Einot-I)Pr(not-IIC)
Therenow aretoo manyunknowns.As a result,the (IV- 1, 1, 1) model fails to
be
identifiable.4
What is the significance of this fact? If you insist that a model must
be identifiable,then this is a reason to favour the (N-1, 1) model over the
(IV- 1, 1, 1) model. If you do not insist on this, you nonethelessmightwantto
comparethe likeliest version of the (N-1, 1) model with any of the likeliest
versionsof the (IV- 1, 1, 1) model,andascertainwhichof themis likelier-i.e.
which confersa higherprobabilityon the observations.The answeris thatthe
models tie. As we've just seen, if the maximumlikelihoodestimateof Pr(EIC)
in the firstmodel sets thatparameterequal to 0.72, the second model can do
just as well, but no better, by choosing values for Pr(E[I),Pr(Einot-I),and
Pr(I[C),so that Pr(EtC)comes out having a value of 0.72.
In summary,the (N-1, 1) model has the advantageof being simpler and
also of being identifiable.Fromthe point of view of accommodatingthe data,
the two models do equally well. We have identifiedno groundon which the
(IV-1, 1, 1) model is betterthanthe (N-1, 1) model.5
3 Multiple causes, multiple effects, and multiple intervening
variables
The models just consideredtreat C, I, and E as dichotomousvanables, but
the result can be generalized. Suppose there are three dichotomous causal
4
To put this point more carefully, the (IV-1, 1, 1) model has five independentlyadjustable
parameters(Pr[C],Pr[IIC],Pr[Itnot-C],Pr[EgI],andPr[Egnot-I]),whereasfrequencyinformation
aboutC andE furnishesonly threeindependentobservations;you observef(E), f(C), f(EgC),and
f(Elnot-C),butthe firstof thesefrequenciesis a functionof theremainingthree.Thisis why (IV-
1, 1, 1) is not identifiable.
5
Of course,one might have backgroundknowledgethatleads one to thinkthatthe intervening-
variablemodel is betterin this instance.However,the focus of the presentpaperis on how the
dataandfeaturesintrinsicto the models consideredpermita choice to be made.This is why I so
farhave restrictedmy attentionto how parsimoniousthe models areandhow probablethey say
the observationsare.
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