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Inferences about unobserved causes in humancontingency learning

To cite this Article: , 'Inferences about unobserved causes in human contingencylearning', The Quarterly Journal of Experimental Psychology, 60:3, 330 - 355To link to this article: DOI: 10.1080/17470210601002470URL: http://dx.doi.org/10.1080/17470210601002470

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Inferences about unobserved causes in humancontingency learning

York Hagmayer and Michael R. WaldmannUniversity of Gottingen, Gottingen, Germany

Estimates of the causal efficacy of an event need to take into account the possible presence and influ-ence of other unobserved causes that might have contributed to the occurrence of the effect. Currenttheoretical approaches deal differently with this problem. Associative theories assume that at least oneunobserved cause is always present. In contrast, causal Bayes net theories (including Power PC theory)hypothesize that unobserved causes may be present or absent. These theories generally assume inde-pendence of different causes of the same event, which greatly simplifies modelling learning and infer-ence. In two experiments participants were requested to learn about the causal relation between asingle cause and an effect by observing their co-occurrence (Experiment 1) or by actively interveningin the cause (Experiment 2). Participants’ assumptions about the presence of an unobserved cause wereassessed either after each learning trial or at the end of the learning phase. The results show an inter-esting dissociation. Whereas there was a tendency to assume interdependence of the causes in theonline judgements during learning, the final judgements tended to be more in the direction of anindependence assumption. Possible explanations and implications of these findings are discussed.

Most events have many causes, most of which wedo not know or cannot observe. Nevertheless,despite the fact that the observed effects are alsoinfluenced by unobserved events, we are capableof learning about the causal efficacy of observedcauses. A large body of theoretical and empiricalknowledge has been accumulated on how wemake causal strength assessments in such situ-ations (Cheng, 1997; Shanks, 1987; Shanks,Holyoak, & Medin, 1996; Waldmann &Hagmayer, 2001; Wasserman, Elek, Chatlosh, &Baker, 1993; White, 2003). However, the question

has largely been neglected in the literature as towhat we can learn about the causes that wecannot observe (but see Kushnir, Gopnik,Schulz, & Danks, 2003; Luhmann & Ahn,2003). For example, in the early days of AIDSnobody knew about the virus and its lethalcapacity. The known observable causes includedminor infections, which apparently led to serious,live threatening conditions, such as pneumonia.However, the seriousness of the disease in lightof the rather harmless causes led to the searchfor additional unobserved causes. Eventually the

Correspondence should be addressed to Y. Hagmayer or M. Waldmann, Department of Psychology, University of

Gottingen, Gosslerstr. 14, 37073 Gottingen, Germany. E-mail: [email protected] or michael.waldmann@

bio.uni-goettingen.de

We thank Tom Beckers, Marc Buehner, and an anonymous reviewer for helpful comments on an earlier draft of this paper. We

also thank M. Rappe for his help with planning and running Experiment 2. Portions of this research were presented at the Twenty-

fifth Annual Conference of the Cognitive Science Society, Chicago, in 2004.

330 # 2007 The Experimental Psychology Society

http://www.psypress.com/qjep DOI:10.1080/17470210601002470

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY

2007, 60 (3), 330–355

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HIV virus was discovered, and theories weredeveloped that traced the complex causal pathwaysunderlying the disease.

Although most theories of causal learning didnot directly address inferences about unobservedcauses, most theories took the possibility of suchcausal factors into account. The question of howto assess causal strength when there are unob-served causes has challenged normative theoriesof causality and psychological theories of causalreasoning for some time. A number of differentaccounts have been proposed analysing how thecausal strength of an observed factor can be accu-rately estimated if certain assumptions are madeabout potential unobserved causes. In this articlewe study more directly what people infer aboutunobserved causes during learning. First we givea brief overview of how associative and causalBayes net theories handle unobserved causes.Based on these theories we derive predictionsabout the inferences that people should makeabout the presence and causal influence of unob-served causes. In the second part of the article wepresent two experiments in which we assessedthe assumptions of learners about the strengthand probability of unobserved causes. In the finalsection we discuss potential theoretical impli-cations of these findings.

Theoretical accounts of unobservedcauses

Unobserved events can be linked in complexnetworks and can simultaneously cause severalobserved events, leading to confoundings (seePearl, 2000; Spirtes, Glymour, & Scheines,1993). In the present research we focus on asimple causal situation consisting of a single obser-vable cause C and one possible unobserved cause A,both influencing a joint observable effect E.Figure 1 illustrates the causal model. The twoobservable events C and E are statisticallyrelated. There are three questions one might askin this kind of situation: How does C influenceE? How are C and A related to each other? Howdoes A influence E?

Associative theoriesAssociative theories, such as the Rescorla–Wagner theory (Rescorla & Wagner, 1972),would model this task as learning about the associ-ation between a cue representing the observedcause and an outcome representing the effect.Along with the observable cause cue a secondbackground (or context) cue would be part of themodel. This background cue is assumed to bealways present and to represent all other factorsthat might also generate the outcome. Thus, thebackground cue would play the role of represent-ing the unobserved cause A in the outlined causalmodel. According to the Rescorla–Wagner rule,only weights of cues that are present in the learn-ing trial are being updated. Therefore, the perma-nently present background cue will generallycompete with the cause cue in cases in which thecause cue is present. If the outcome is alsopresent, the associative weights of both cues willbe raised; if the outcome is absent, the weightswill be lowered. However, in cases in which thecause cue is absent, only the weight of the back-ground cue will be altered. At the asymptote oflearning the associative weight of the observedcause will equal the contingency of the cause cueand the outcome, which is defined as thenumeric difference between the probabilities ofthe effect in the presence and the absence of theobserved cause—that is, DP ¼ P(ejc) – P(ej�c);(see Cheng, 1997; Danks, 2003). The associativeweight of the background cue will correspond tothe probability of the outcome in the absence of

Figure 1. Causal model of a situation in which an observable cause

C and an unobserved cause A both influence an observable effect E.

The question marks indicate the relations that participants were

requested to assess.

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2007, 60 (3) 331

INFERRING UNOBSERVED CAUSES

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the cause cue. Thus, the more often the outcome(¼ effect) occurs on its own, the higher the associ-ative weight of the background cue will be.

Consequently associative theories wouldpredict that participants assume an unobservedcause to be always present. In addition, these the-ories predict that at the asymptote the unobservedcause’s strength would match the probability of theeffect in the observable cause’s absence, which cor-responds to the associative weight linking thecontext cue and the effect, and that the estimatedcausal influence of the observable cause corre-sponds to the observed contingency.

Causal Bayes netsCausal Bayes net theories of causal learning offeraccounts for a large variety of problems—forexample, for hypothesis testing, model generation,and parameter estimation (see Danks, in press;Glymour, 2001; Griffiths & Tenenbaum, 2005;Spirtes et al., 1993; Tenenbaum & Griffiths,2003). In fact, causal Bayes net theories are arather heterogeneous group of theories. While alltheories agree on the representation of causalstructures as acyclic graphs capturing relations ofconditional dependence among the events, theyassume different processes of learning and infer-ence. Whereas some theories rely on Bayesianinferences involving the application of Bayes’theorem (e.g., Tenenbaum & Griffiths, 2003),other theories do not (e.g., Cheng’s, 1997, PowerPC theory; see below).

All causal Bayes net theories agree on theassumption that causal learning and reasoningare based on structured causal models (i.e., causalBayes nets) of a domain. Causal Bayes netsconsist of graphs of causal models and parametersrepresenting base rates, causal strength, and causalinteractions. The graph depicted in Figure 1depicts a common-effect structure in which twocauses—one observable and one unobserved—affect a single effect. The graphs specify relationsof conditional dependence and independenceamongst the variables within the model. Forexample, a common-effect model such as the onedepicted in Figure 1 represents the assumptionthat the observed cause and the unobserved cause

are independent of each other, that the two causesare related to the effect, and that the two causesare negatively related conditional upon theeffect—that is, if the effect and one of the causesis observed the presence of the second cause is lesslikely than its presence (i.e., explaining away).Thus, causal Bayes net theories in general assumethat different causes of a common effect are inde-pendent of each other, as long as there is nofurther causal factor that affects the two causesand thereby creates dependence between them.

Power PC theory. Cheng’s (1997) Power PC analy-sis of the causal impact of a single cause can beviewed as a special case of a causal Bayes net inwhich two causes independently influence a jointcommon effect (see Glymour, 2001). Based onthis theory some predictions for the probabilityand causal power of an unobserved cause can beinferred. The theory presupposes that the occur-rence of the effect E is a consequence of thecausal powers of the observed cause C (pc), of anunobserved cause A (pa), and of their base ratesP(c) and P(a). Formally the probability of theeffect equals the sum of the base rates of the twocauses multiplied by their causal power minusthe intersection of the causes multiplied by bothcausal powers:

P(e) ¼ P(c) � pc þ P(a) � pa � P(c) � P(a) � pc � pa:

Therefore the probability of the effect E con-ditional upon an observation of cause C— thatis, P(c) ¼ 1—is

P(ejc) ¼ pc þ P(ajc) � pa � P(ajc) � pc � pa 1

The probability of the effect given that theobservable cause C is absent—that is, P(c) ¼ 0—is

P(ej � c) ¼ P(aj � c) � pa 2

Equations 1 and 2 hold irrespective of whetherthe observed cause C and the unobserved cause Aare dependent, P(ajc) = P(aj�c), or independent,P(ajc)¼ P(aj�c). Both Equation 1 and Equation 2

332 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2007, 60 (3)

HAGMAYER AND WALDMANN

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have at least two parameters that cannot be directlyestimated on the basis of the observed relationbetween C and E. Thus, neither the probabilitynor the causal power of the unobserved cause canbe directly inferred from the observable data.

Power PC theory makes the additional assump-tion that different causes of the same effect areindependent of each other—that is, the theoryassumes that the observed and the unobservedcauses are not statistically related, P(ajc) ¼P(aj�c) ¼ P(a). This assumption allows it toderive estimates for the causal power of the obser-vable causes even if there are other unknown causesgenerating the same effect. Based on this assump-tion Equations 1 and 2 can be simplified into

P(ejc) ¼ pc þ P(a) � pa � P(a) � pa � pc 10

The probability of the effect given that theobservable cause C is absent—that is, P(c) ¼ 0—is

P(ej � c) ¼ P(a) � pa 20

Inserting 20 into 10 yields

P(ejc) ¼ pc þ P(ej � c)� P(ej � c) � pc 100

which can be rearranged into the well-knownformula for generative causal power

pc ¼P(ejc)� P(ej � c)

(1� P(ej � c): 3; power formula

What inferences can be made about the unob-served cause? The independence assumption ofPower PC theory implies that the probability ofthe unobserved cause stays the same regardless ofwhether the observed cause has occurred or not.Moreover, Equation 20 implies that the causalpower of the unobserved cause and its probabilityhave to be at least as big as the probability of theeffect in the absence of the observed cause, pa �

P(ej�c) and P(a) � P(ej�c). Equations 10 and 20

provide additional constraints for the twounknown parameters. Equation 20 requires thatthe product of the estimated probability and the

causal power of the unobserved cause has to equalthe observable probability of the effect in the obser-vable cause’s absence. In sum, Power PC theorylike other causal Bayes net theories assumes inde-pendence between causes and provides constraintsfor the probability and causal power of unobservedcauses. This assumption is necessary to makeprecise inferences about the power of observedvariables. However, these theories may also dropthis assumption if it seems unwarranted. In thenext section we give some examples for inferencesthat are possible without assuming independence.

Inferring unobserved causes

In this section we present two formal Bayesiananalyses of potential inferences about an unob-served cause, which do not rely on the assumptionof independence. The first analysis is based onprobability calculus and concerns inferencesabout the probability and strength of a unobservedcause based on a set of observed data. The secondanalysis deals with inferences based on singleobservations of a given cause and an effect. Thetwo analyses show that observable causal relationsnormatively allow some inferences to be drawnabout unobserved causes, even if no independenceamong the causes is assumed. Readers not inter-ested in the formal details may prefer to skip thissection. The results of the analyses are summarizedin the next section.

The first analysis is based on the causal modeldepicted in Figure 1. Two causes, C and A, oneof which is not observable (A), are assumed toinfluence a joint effect E. Based on the chainrule of probability theory, the joint probabilitiesof the three events can be factorized as follows:

P(E, A, C) ¼ P(EjA, C)�P(A, C)

While the joint probability of the two observableevents P(E, C) can be estimated on the basisof the observable events, neither the probabilityof the effect conditional on the pattern of theobserved and unobserved causes, P(EjA,C), northe joint probability of the observed and



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unobserved cause, P(A, C), can be estimated. Thus,additional assumptions have to be made. There isone popular assumption that is frequently madewith causal Bayes nets (Jensen, 1997; Pearl,1988), which is the so called noisy-or assumption.This assumption informally states (a) that eachcause deterministically causes the effect unless itsimpact is intercepted by an inhibitor, and (b)that the inhibitors are independent of each other.Note that it is not assumed that the causes areindependent. For example, if there are twocauses of a common effect, and both are present,then either the first cause generates the effect orthe second cause generates the effect or both do.Thus, the effect will only fail to occur if bothcauses are inhibited. Formally

P(eja:c) ¼ 1� (1� P(ej�a:c)) � (1� P(eja:�c))

Based on the noisy-or rule the joint probability ofcause and effect co-occurring can be calculated by

P(e:c) ¼ P(eja:c) � P(ajc) � P(c)

þ P(ej�a:c) � P(�ajc) � P(c)

¼ {1� ½1� P(ej�a:c)� � ½1� P(eja:�c)�}

� P(ajc) � P(c)þ P(ej�a:c) � P(�ajc) � P(c)

Conditionalized on the presence of the observablecause C, the following equation results:

P(ejc) ¼ {½1� ½1� P(ej�a:c)�

� ½1� P(eja:�c)�} � P(ajc)

þ P(ej�a:c) � P(�ajc)

4

The probability of the effect occurring on its owncan also be calculated:

P(e:�c) ¼ P(eja:�c) � P(aj�c) � P(�c)

þ P(ej�a:�c) � P(�aj�c) � P(�c)

According to the noisy-or assumption absentcauses never lead to the effect—that is, P(ej�

a.�c) ¼ 0. Therefore

P(e:�c) ¼ P(eja:�c) � P(aj�c) � P(�c)

Conditional on the absence of the observable causeC, this equation can be further simplified into

P(ej�c) ¼ P(eja:�c) � P(aj�c) 5

It should be noted that thus far no assumptionabout the statistical relation between the observa-ble cause and the unobserved cause has been made.However, Equations 4 and 5 also show that noprecise inferences about the probability andcausal strength of the unobserved cause can bedrawn on the basis of the observable eventsalone. There are at least two unknown parametersin each of these equations, which refer to the con-ditional probabilities that involve the unobservablecause. Nevertheless, Equations 4 and 5 do provideconstraints for the possible causal strength andprobability of the unobserved cause. In particularthe probability of the effect in the absence of theobserved cause provides a rather simple constraint.As Equation 5 shows, this probability represents alower boundary for the probability of the unob-served cause in the observable cause’s absence,P(a j�c) � P(ej�c), and for the probability thatthis cause generates the observable effect on itsown, which corresponds to its causal power,P(eja.�c) � P(ej�c)). Thus, even if no preciseinferences about the parameters of unobservedcauses can be drawn this analysis shows thatthere are numeric boundaries for plausible infer-ences. The more often the effect occurs on itsown, the more often the unobserved cause hasprobably been present and the stronger its causalimpact upon the visible effect will be.

The second analysis shows that at least approxi-mate inferences about the presence of the unob-served cause A can be drawn by observing thepresence or absence of the cause C and effect Eon individual trials. There are four possible pat-terns of events that might be observed: cause Cand effect E both being present (e.c), cause Cbeing present without the effect (�e.c), the effectbeing present without the observable cause



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(e.�c), and both events being absent (�e.�c).From a Bayesian perspective the question iswhether it is more likely for each of these fourpossible observations that an unobserved cause Ais present (a) or that it is absent (�a). UsingBayes’ theorem the probability of the unobservedcause being present can be calculated as

P(AjE, C) ¼P(EjA, C) � P(AjC)

P(EjC):

Given that the presence of both cause C and theeffect E are being observed, the posterior oddsfor an unobserved cause to be present can be calcu-lated by

P(aje:c)

P(�aje:c)¼

P(eja:c)

P(ej�a:c)�

P(ajc)

P(�ajc)6

(i.e., posterior odds ¼ likelihood ratio �

prior odds)

Let us first consider the likelihood ratio: Therelevant question is whether the probability ofthe effect is higher given both the unobservedand the observed cause are present or highergiven only the observable cause is present—thatis, whether P(eja.c) . P(ej�a.c). If the unobservedcause is assumed to have a positive causal influenceupon the effect, then the effect is more likely ifboth causes are present than if only one cause ispresent, P(eja.c) . P(ej�a.c). Therefore, an obser-vation of both the cause and the effect makes thepresence of an unobserved cause more likely.Nevertheless, no definite prediction is possiblewithout taking into account the prior odds. Aslong as P(ajc) is equal or bigger than P(�ajc), theprior odds are larger than or equal to one, whichimplies that the presence of the unobserved causeis more likely than its absence. This means thatas long as we have reason to assume that the pre-sence of an unobserved cause is a priori equallylikely or more likely than its absence, its presenceshould be predicted if a cause and its effect areboth being observed.

We now turn to the second possible pattern ofevents. If only the cause is observed without the

effect then the posterior odds of the unobservedcause are given by

P(aj�e:c)

P(�aj�e:c)¼

P(�eja:c)

P(�ej�a:c)�

P(ajc)

P(�ajc)7

If both causes have a positive causal influencethen P(eja.c) . P(ej�a.c) and therefore P(�eja.c) , P(�ej�a.c). Thus, an observation of acause without its effect makes the presence of anunobserved cause less likely. Again a guess aboutthe prior odds is needed to arrive at a definitejudgement.

However, it is important to note that the priorodds are identical in all cases in which the cause isobserved. Regardless of whether e.c or�e.c is beingobserved, the prior odds are P(ajc) / P(�ajc) (seeEquations 6 and 7). This implies that the posteriorodds of the unobserved cause are higher in cases inwhich both cause and effect are observed than incases in which only the cause is observed nomatter what the prior odds are. Thus, even if nodefinite estimates are possible without specificassumptions about the prior probabilities of theunobserved cause, the posterior odds should differ.

If the effect is observed without the observablecause, the posterior odds are calculated as

P(aje:�c)

P(�aje:�c)¼

P(eja:�c)

P(ej�a:�c)�

P(aj�c)

P(�aj�c)8

Thus, as long as the unobserved cause has aspositive influence, P(ej a.�c) . P(ej�a.� c), thisobservation makes the presence of the unobservedcause more likely. However, an even more distinctprediction is possible based on the often-statedcausal principle that nothing happens without acause (“nihil fit sine causa”, Audi, 1995). Basedon this principle, P(ej�a.�c) should equal zero,which implies that the unobserved cause has tobe present for sure. Thus, an observation of aneffect without any visible cause makes the presenceof an unobserved cause necessary.

Finally, if neither the cause nor its effect ispresent, the posterior odds of the unobserved



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cause are

P(aj�e:�c)

P(�aj�e:�c)¼

P(�eja:�c)

P(�ej�a:�c)

�P(aj�c)

P(�aj�c)

9

If the effect never happens without any cause thenP(�ej�a.�c) ¼ 1. Thus, an observation of neithercause nor effect implies that it is less likely that theunobserved cause has happened. Again, for a defi-nite prediction assumptions about the prior oddshave to be made. However, as in the cases inwhich the observable cause is present, the priorodds are the same in all cases in which the obser-vable cause is absent (see Equations 8 and 9). Theprior odds of the unobserved cause given an obser-vation of eithere.� c or �e.�c are P(aj�c) /P(�aj�c). This implies that the posterior odds ofthe unobserved cause are lower when none of theevents is observed than when the effect is observedby itself.

How can the prior odds be estimated? As wehave shown in the first analysis the observableprobability of the effect in the cause’s absence pro-vides a lower boundary for the probability of theunobserved cause, P(aj�c) ¼ 1 – P(�aj�c) �P(ej�c). Thus, after observing several trials atleast the prior odds for cases in which theobservable cause is absent can be estimated fromthe data.

The two analyses provided in this section showthat even if no assumption of independence amongthe observed and the unobserved causes of acommon effect is made, it is possible to derivesome approximate inferences about the unob-served cause. This is true for individual trials aswell as for complete data sets. Thus, informedguesses about unobserved causes can be made.For these analyses only probability calculus andBayes’ theorem were used. Consequently causalBayes net theories can be adapted to cases inwhich the assumption of independence of causesseems unwarranted.

SummaryCurrent theories of causal learning consider unob-served causes and make assumptions that allow itto make inferences about these causes.Associative theories assume that an unobservedcause (i.e., the constant background) is alwayspresent and therefore independent of observablecauses. Causal Bayes net theories including thespecial case of Power PC theory typically assumethat different causes are independent of eachother. This assumption allows these theories tomake precise estimates about unobserved causes.However, despite the fact that these theories ingeneral assume independence among differentcauses of a joint effect, these theories can alsomodel situations in which the independenceassumption does not hold. The analyses in the pre-vious section have shown that in the case of depen-dence between observable and unobservable causes(i.e., confounding) it is normatively not possible toderive precise quantitative inferences about theprobability and strength of the unobservedcauses. Nevertheless it is possible to use Bayesianinferences to provide reasonable estimates. Evenif no assumptions about dependence are made,the probability of the effect in the absence of thecause marks a lower boundary for the probabilityof the unobserved cause in the observable cause’sabsence, P(aj�c) � P(ej�c).

Associative as well as causal Bayes net theoriesagree that P(ej�c) is to a certain extent indicativeof the causal strength of the unobserved cause.But whereas associative theories generally regardthis probability as a valid indicator, Power PCand other causal Bayes net theories view this con-ditional probability as a lower boundary of thecausal strength of the unobserved cause.

Causal Bayes net theories are also able to makespecific predictions for different patterns of obser-vations by using Bayesian inferences about unob-served causes (see previous section for details). If acause and an effect are both observed it is morelikely that an unobserved second cause is alsopresent than when the cause is observed withoutan effect, P(aje.c) � P(aj�e.c). The existence andpresence of an unobserved cause is certain if aneffect is observed without any visible cause,



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P(aje.�c)¼ 1. If neither cause nor effect is observed,the presence of an unobserved cause is unlikely.

Experimental evidence

A number of researchers have provided evidencethat people can and do infer the existence of unob-served causes. For example, children and adultsinfer the unobserved mental causes (e.g., desiresand beliefs) of other people’s actions (Gopnik &Wellman, 1994; Wellman, 1990). People alsoassume that there is an essential unobservedentity in categories, which has been speculated toplay the role of a unobserved common cause ofthe observable features of category members(Gelman, 2003; Medin & Ortony, 1989; Rehder,2003; Rehder & Hastie, 2001). In scientificreasoning people often infer unobserved entitiesto account for the observed events (Gopnik &Meltzoff, 1997; Keinath & Krems, 1999; Krems& Johnson, 1995). An example for recent studieson unobserved causes is a series of experimentsconducted by Gopnik and colleagues (Gopniket al., 2004; Kushnir et al., 2003). In one of theirstudies children were presented a stick-ballmachine where they saw two balls moving togetherup and down for four times. Next an experimentermoved each ball by hand, which did not have anyvisible effect on the other ball. Children wereasked to give an explanation. In accordance withcausal Bayes net theories children as young as 41

2years were able to reason that neither ball is thecause of the other so that there must have beenan unobserved common cause that makes thetwo balls move together. These and other resultsshow that children (and adults) are able to cor-rectly infer the presence of an unobserved causefrom a mixture of observations and interventions.

Our focus differs from these studies, however.Whereas Gopnik and colleagues (2004) weremainly interested in the induction of the presenceof an unobserved common cause of multipleeffects, our goal is to study inferences about theprobability and the causal strength of unobservedcauses that are known to exist. Moreover,whereas previous studies have focused on unob-served common causes of multiple observed

effects, our experiments explore inferences aboutcommon-effect structures in which one cause isunobserved.

To the best of our knowledge only one study hasthus far investigated some of these questions.Luhmann and Ahn (2003) have confronted par-ticipants with a single causal relation consistingof one observable cause and one observable effect(see also Ahn, Marsh, & Luhmann, in press).They showed that many participants were willingto judge the causal strength of unobserved causesthat they had never observed during causal learn-ing. In their first experiment Ahn and colleaguesalso found that the estimated causal influence ofthe unobserved causes declined when theirnumber increased (the observable causal relationwas kept constant). In a second study Luhmannand Ahn manipulated the probability of the obser-vable effect when the observable cause was absent.It turned out that participants judged the causalstrength of a single unobserved cause to behigher if P(ej�c) was .5 than if it was zero. Thisfinding is consistent with the theoretical accountsintroduced above. Unfortunately, no estimates forthe dependency between the observable and theunobserved cause were collected in these exper-iments. Therefore it is not possible to differentiatebetween an associative and a Bayesian account.

In an unpublished experiment (summarized inAhn et al., in press) participants’ inferences aboutthe unobserved cause’s presence were measuredon each learning trial. It turned out that partici-pants assumed the unobserved cause to bepresent when the effect had occurred on its ownand to be absent when neither the visible causenor the effect had occurred. This finding is inaccordance with the predictions derived from aBayesian analysis.

With the experiments presented in this paperwe intended to go beyond these previous findings.In addition to causal strength estimates, we alsocollected assessments of the probability of theunobserved cause in the presence and absence ofthe observed cause and effect. We used a varietyof methods to measure people’s assumptionsabout the interdependence between observed andunobserved causes in different task contexts.



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OUTLINE OF EXPERIMENTS

The aim of the following two experiments was toexplore what assumptions participants makeabout the presence and causal strength of an unob-served cause in a trial-by-trial learning task and totest whether these assumptions conform to thepredictions of any of the discussed theoreticalmodels. In both experiments participants learnedabout the causal relation between an observablecause and an effect. In addition, participantswere told that there was one other possible butunobserved cause of the effect. Participants neverreceived any further information about the unob-served cause beyond this hint.

The statistical relation between the observablecause and the effect was manipulated in two exper-iments. In Experiment 1 the contingency betweenthe observable cause and the effect was kept con-stant while the causal power varied across con-ditions. In Experiment 2 causal power was keptconstant while the contingency varied. We usedthis manipulation to investigate whether partici-pants would base their estimates of causal strengthon the contingency between cause and effect or onthe causal power of the cause, which could beinferred from the observations if independenceamongst causes is assumed. Previous researchyielded mixed evidence about this issue. Whilesome researchers found support for the use ofcausal power (Buehner & Cheng, 1997; Buehner,Cheng, & Clifford, 2003) others found supportfor the use of contingencies (Lober & Shanks,2000). Since we were interested in causal strengthestimates, we were interested in finding outwhether learners are sensitive to contingencies orcausal power when estimating causal strength.

The manipulation of the statistical relationbetween the observable cause and the effectshould also affect participants’ guesses about theunobserved cause’s strength. As we have outlinedin the Introduction, the probability of the effectin the observable cause’s absence is an indicatorof the unobserved cause’s strength and provides alower boundary. Thus, the estimated causalstrength should rise in parallel to the probabilityof the effect in the absence of the observed cause.

We also manipulated whether participantscould only passively observe the cause and theeffect (Experiment 1), or whether they couldactively intervene and decide when the causeshould be present or absent (Experiment 2).Mere observations without interventions do notpermit it to make strong assumptions about theinterdependence between the alternative causes.Either is possible, dependence or independence.Thus, participants may assume that the observedand the unobserved causes are dependent. Incontrast, interventions should typically generateindependence. Since the presence or absenceof unobserved causes is unknown, deliberatelysetting a cause normally creates independencebetween the two causes (see Pearl, 2000; Spirteset al., 1993; Woodward, 2003, for theoreticalanalyses of interventions). Therefore participantsshould assume independence (see Gopnik et al.,2004; Hagmayer, Sloman, Lagnado, &Waldmann, in press; Sloman & Lagnado, 2005;Waldmann & Hagmayer, 2005, for psychologicalevidence).

We used two different tasks to assess partici-pants’ estimates of the probability of the unob-served cause: First, participants were asked toguess on each learning trial whether the unob-served cause was present or absent. Then, at theend of the learning session we requested additionalsummary estimates. Both measures were used toassess participants’ assumptions about indepen-dence between observed and unobserved causes.By making several predictions in a series of trialsparticipants generate patterns of alternativecauses, which allow analysis of the dependencebetween the causes that participants implicitlygenerate. It is known in the literature on learningthat participants tend to match the probability ofevents when generating sequences (see reviews byHernstein, 1997; Myers, 1976; Vulkan, 2000).Therefore we hypothesized that participants’predictions would mirror their assumptions aboutthe probability of the unobserved cause.Participants’ final overall estimates are more expli-cit; they are based on summary information storedin memory. We were interested in whether thesetwo types of measures would yield similar



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estimates, or whether they would diverge. Sincethe online predictions are based on single cases, itmay well be that these estimates focus on differentaspects of the data from those focused on by thesummary statements (see Catena, Maldonado, &Candido, 1998, for an example of such a dis-sociation in the domain of causal strength esti-mation). For causal strength assessmentssummary estimates are, in our view, the more rel-evant evidence for independence assumptions asboth are collected at the end of the learning phase.

EXPERIMENT 1

Participants’ task in the first experiment was tolearn about the causal relation between a microbeinfecting flowering plants and the discolorationof their bloom. In addition, participants wererequested to draw inferences about the presenceand causal influence of a second but unobservedtype of microbe, which may also affect the plant’scoloration. This first experiment investigatedobservational learning—that is, participantscould only observe whether a flower was infectedand discoloured; they were not able to intervenethemselves. Two experimental factors weremanipulated: The first factor was the strength ofthe causal relation between the observablemicrobe and the discoloration of three types ofplants. This factor was manipulated within sub-jects in a counterbalanced order. Contingenciesbetween infection and discoloration were keptconstant, but the two conditional probabilitiesP(ejc) and P(ej�c) were increased across con-ditions. This increase has two implications: (a) itimplies an increasing causal power of the observa-ble cause, and (b) it implies a higher probability orstronger causal strength of the second unobservedmicrobe because the probability of discoloration inthe absence of an infection by the observablemicrobe increased.

As a second factor, which was manipulatedbetween subjects, the learning procedure wasmanipulated: In two experimental conditions par-ticipants were requested to make predictions aboutthe unobserved cause on each learning trial. In one

of these conditions (“prediction after effect”) par-ticipants received information about the presenceof the first microbe and the effect and then hadto predict the presence of the unobservedmicrobe. This information should allow them tomake more informed guesses about the unobservedcause (see the analysis in the section on inferringunobserved causes). Summarizing our analyses inthe Introduction, participants should be quitesure that the unobserved microbe is present if theobservable microbe is absent but the flower is dis-coloured. They should also feel confident that theunobserved cause is absent if neither the observa-ble cause nor the effect has occurred. If causeand effect were both present, learners shouldpredict the unobserved cause to be present moreoften than if the cause was present but the effectwas missing. No feedback was provided aboutthe unobserved cause.

In the second experimental condition (“predic-tion before effect”) participants had to make theirguess after observing the presence or absence of theobservable cause, but before being informed aboutthe occurrence of the effect. Thus, they were firstinformed about the presence or absence of thecause in each trial, and then they were asked toguess whether the unobserved cause was presentwithout receiving feedback about the alternativeunobserved cause. Only after they had madetheir prediction were they informed whether theeffect had occurred at this particular trial or not.In contrast to the “prediction after effect” con-dition this condition does not allow specific infer-ences to be made about the unobserved cause’spresence on an individual trial. Participants’ infer-ences of the unobserved cause prior to effect infor-mation can only be guesses based on observedfrequencies of the effect in the absence of theobservable cause in past trials. As we have outlinedabove, the probability of the effect in the absenceof the cause provides a lower boundary of thebase rate of the unobserved cause. Therefore par-ticipants should use the trials in which theobserved cause is absent to infer the probabilityof the unobserved cause. Assuming independencethey should adapt their predictions for the trialsin which the observed cause is present to this



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probability. If they did that, they would generateindependence between the causes. A thirdcontrol condition presented the learning datawithout requesting trial-by-trial predictions.

After observing the causal relation participantswere requested to rate the causal strength of theobserved as well as of the unobserved microbe.They were asked to estimate the probability ofthe second microbe being present, when the firstmicrobe was present and when the first microbewas absent. Both participants’ online inferencesduring learning and their final estimates wereused to calculate their assumptions about thedependence between the two alternative causes.

Method

Participants and designA total of 36 students from the University ofGottingen, Germany, participated in the exper-iment and were randomly assigned to one of thethree learning conditions.

Materials and procedureParticipants were first instructed to learn about thecausal relation between a fictitious microbe (“col-orophages”) and the discoloration of flowers.They were also told that there was only oneother possible cause of the effect, an infectionwith another fictitious microbe (“mamococcus”),which was currently not observable. Next partici-pants were presented a stack of index cards provid-ing information about individual flowers. Thefront side of each index card showed whether theflower was infected by colorophages or not, andthe backside informed about whether the flowerwas discoloured or not. Then participants wereinstructed about the specific learning procedurein their condition. In the “prediction beforeeffect” condition participants were first shownthe front side of the card, then they had to guesswhether the flower was also infected by the otherunobserved microbe, and finally the card wasturned around by the experimenter revealingwhether the flower was in fact discoloured ornot. In contrast, in the second experimental

condition (“prediction after effect”) the card wasturned around after the presentation of the frontside revealing the presence or absence of discolor-ation. Then the participants were asked to maketheir guesses about the unobserved cause.Guesses were secretly recorded without givingany feedback. In the third, the control, conditionthe cards were simply shown and turned aroundby the experimenter. No inferences about theunobserved cause were requested.

After each learning phase participants wereasked to rate the strength of the causal influenceof the observed and the unobserved cause on ascale ranging from 0 (“no causal influence”) to100 (“deterministic causal influence, i.e., thecause always yields the effect”). Participants werealso asked to estimate how many of 10 flowersthat were infected with the observed microbewere also infected with the other microbe, andhow many of 10 flowers that were not infectedwith the observed microbe were instead infectedwith the other microbe. No feedback was providedabout these assessments.

Three data sets were constructed in a mannerthat the contingency DP was held constant acrossall sets, whereas both P(ej�c) and causal powerwere rising. Each data set consisted of 20 cases.The data sets and their statistical properties areshown in Table 1. All three data sets were shownto every participant in a within-subjects design.Different data sets were introduced as data fromdifferent species of flowers. It was pointed out toparticipants that the effectiveness of the microbesmight vary depending on the species. The orderof the presented data sets was counterbalanced.

Table 1. Data sets shown in Experiment 1 and their statistical

properties

Data set Data set

Observations 1 2 3 Statistics 1 2 3

e.c 6 8 10 P(c) .50 .50 .50

�e.c 4 2 0 P(ejc) .60 .80 1.0

e.�c 1 3 5 P(ej�c) .10 .30 .50

�e.�c 9 7 5 DP .50 .50 .50

Sum 20 20 20 Power pc .56 .71 1.0



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Results

First the estimated causal influence of the observedand the unobserved causes was analysed. Table 2shows the mean ratings. Overall the descriptivemeans indicate increasing estimates for bothcauses. Separate analyses of variance for eachcause with learning condition (before effect, aftereffect, control) as a between-subjects factor anddata set (1, 2, 3) as a within-subjects factor wereconducted. The analysis for the observed causerevealed a significant increase in causal strengthratings, F(2, 66) ¼ 12.7, MSE ¼ 296.6, p , .01,while the other main effect and the interactionfailed to reach significance (Fs , 1). Thispattern of results supports the predictions ofPower PC theory, as the contingency betweenthe cause and the effect remained constant acrossconditions. The analysis for the unobserved causealso resulted in a significant main effect of thefactor data set, F(2, 66) ¼ 4.92, MSE ¼ 408.1,p , .05, which indicated that with increasingP(ej�c) participants tended to assume a strongercausal strength of the unobserved cause. Thisresult conforms to the predictions of all theoreticalaccounts discussed in the Introduction.

The interaction between data sets and learningcondition also turned out to be significant,

F(4, 66) ¼ 4.55, MSE ¼ 408.1, p , .05. Theobserved increase was strongest in the “predictionafter effect” condition followed by the control con-dition. This interaction can be explained by thedifferent affordances of the three learning con-ditions. In the “prediction after effect” conditionparticipants were sensitized to the possible pre-sence and causal strength of the unobservedcause more than in the other two conditions. Aswe have shown in the Introduction, observingboth cause and effect in a single trial allowsinformed predictions regarding the unobservedcause to be derived. No such predictions werepossible in the other two conditions. In the “pre-diction before effect” condition participantslacked the information necessary to make an infer-ence, which might have signalled to them that it isimpossible to draw any inferences. In the controlcondition, they were never asked about the unob-served cause during learning.

Secondly, participants’ implicit assumptionsabout the dependence between the causes wereanalysed. The trial-by-trial predictions of theunobserved cause in the presence and absence ofthe target cause were transformed into conditionalfrequencies and combined into subjective contin-gencies, DP ¼ P(ajc) – P(aj�c).1 This procedure

Table 2. Results of Experiment 1: Mean ratings of causal influence for observed and unobserved causes

Causal strength: Observed cause Causal strength: Unobserved cause

Condition Data Set 1 Data Set 2 Data Set 3 Data Set 1 Data Set 2 Data Set 3

“Before effect” 57.5 (22.2) 68.3 (15.9) 78.3 (19.9) 50.5 (30.7) 48.3 (23.7) 46.7 (27.1)

“After effect” 64.1 (29.1) 70.8 (21.1) 77.9 (23.7) 34.2 (22.3) 61.7 (33.0) 73.8 (27.1)

Control 57.5 (19.1) 75.8 (12.4) 84.2 (16.2) 43.3 (25.6) 38.3 (21.7) 54.2 (19.3)

Note: Standard deviations in parentheses.

1 We used the following formulae to derive Pgen(ajc) and Pgen(aj�c) from the observed frequencies with which different patterns of

events were predicted:

Pgen(ajc) ¼ ½Pgen(a.e.c)þ Pgen(a.�e.c)�=

½Pgen(a.e.c)þ Pgen(�a.e.c)þ Pgen(a.�e.c)þ Pgen(�a.�e.c)�

Pgen(aj�c) ¼ ½Pgen(a.e.�c)þ Pgen(a.�e.�c)�=

½Pgen(a.e.�c)þ Pgen(�a.e.�c)þ Pgen(a.�e.�c)þ Pgen(�a.�e.�c)�



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was based on the assumption that participants’predictions would reflect their underlying prob-ability estimates due to probability matching. Onthe left side of Table 3 the mean contingenciesthat were generated online are listed.Participants’ final ratings of the conditional fre-quency of the unobserved cause in the presenceand absence of the observed cause were also trans-formed into subjective contingencies. Means areshown on the right hand side of Table 3. An analy-sis of variance of the generated contingencies withthe between-subjects factor “learning procedure”and the within-subjects factor “data set” yielded asignificant trend from positive to negative assess-ments, F(2, 44) ¼ 22.1, MSE ¼ 749.8, p , .01.No other main effect or interaction was found.Thus, the negative trend proved to be independentof the learning condition. The estimated contin-gencies showed a similar negative trend, F(2, 66)¼ 4.2, MSE ¼ 753.9, p , .05. They also differedslightly across learning conditions, F(2, 33) ¼ 2.8,MSE¼ 1,646.4, p , .10, but the interaction failedto reach significance.

To follow up these results more closely, gener-ated and estimated contingencies were comparedto an assumption of independence—that is, theempirically obtained contingencies were testedfor deviations from zero using one-sample t tests.The analyses showed that the contingencies gener-ated online were significantly above zero if P(ej�c)was .1 (Data Set 1) and significantly below zero ifP(ej�c) was .5 (Data Set 3).

The pattern for the final summary estimatesdiffered from the pattern for the generated depen-dencies. In comparison to an assumption of inde-pendence, only marginally significant deviations

were found for most of the estimates. The esti-mates for Data Set 1 in the “prediction beforeeffect” condition were the only exception. Theserather small deviations are consistent with anassumption of independence. This result pointsto a dissociation between online and post hocassessments. While participants generated a posi-tive dependence when the effect had neveroccurred in the absence of the observed causeand a negative dependence when it had occurredfairly often in the absence of the observablecause, the final summary estimates hoveredaround independence.

In a third analytical step the predictions regard-ing the unobserved cause, which participants madeonline on individual trials, were analysed. For thisanalysis the patterns of events generated in theonline judgements were transformed into subjec-tive probabilities of the unobserved cause con-ditional upon the patterns of the observed events.This means that they were transformed into prob-abilities of A conditional on both cause C andeffect E, Pgen(AjC, E), in the “prediction aftereffect” condition and into probabilities of A con-ditional on C, Pgen(AjC), in the “predictionbefore effect” condition. Table 4 shows the meanconditional probabilities derived from the gener-ated patterns. Note that in Data Set 3 participantsnever observed the effect to be absent when theobservable cause was present. The probabilities inthe “prediction after effect” condition conformedfairly well to the theoretical predictions derivedin the Introduction. Participants judged the unob-served cause to be much more likely when theeffect had occurred in the observable cause’sabsence than when neither cause nor effect had

Table 3. Results of Experiment 1: Mean generated and estimated dependencies of observed and unobserved causess

Generated dependence Estimated dependence


“Before effect” .33 (.377) .08 (.549) –.22 (.273) .30 (.252) .06 (.360) –.03 (.337)

“After effect” .17 (.237) .18 (.396) –.29 (.204) –.01 (.318) .00 (.341) –.23 (.470)

Control — — — .14 (.271) .12 (.269) .13 (.234)

Note: Values represent contingencies. Standard deviations in parentheses.



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Table 4. Results of Experiment 1: Mean probabilities of unobserved cause generated by participants conditional upon each pattern of observations

“Before effect” condition

Data Set 1 Data Set 2 Data Set 3

Pgen(ajc) Pgen(aj�c) Pgen(ajc) Pgen(aj�c) Pgen(ajc) Pgen(aj�c)

.66 (.22) .32 (.20) .50 (.35) .53 (.26) .27 (.16) .48 (.21)

“After effect” condition


Pgen(aje.c) Pgen(aj�e.c) Pgen(aje.�c) Pgen(aj�e.�c) Pgen(aje.c) Pgen(aj�e.c) Pgen(aje.�c) Pgen(aj�e.�c) Pgen(aje.c) Pgen(aj�e.c) Pgen(aje.�c) Pgen(aj�e.�c)

.40 (.34) .21 (.35) .92 (.29) .07 (.12) .52 (.43) .29 (.40) .92 (.15) .05 (.16) .44 (.38) — 1.0 (0.0) .02 (.06)


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occurred. In fact, participants in this conditionwere almost certain that the unobserved causewas present, when the observable cause wasabsent, Pgen(aj�c.e) . .90, and absent otherwise,Pgen(aj�c.�e) , .10. An analysis of variance withthe two within-subjects factors “data set” (1, 2,3) and “presence of effect” (present vs. absent)yielded a highly significant main effect of thefactor involving the effect’s presence, F(1, 11) ¼347.0, MSE ¼ 0.042, p , .01. No other effectturned out to be significant. Participants’ predic-tions were less clear cut when the observablecause was present. At least the descriptive differ-ences pointed in the right direction. Participantsconsidered the unobserved cause to be morelikely when the observable cause and the effecthad occurred than when only the observablecause had occurred without the effect. However,an analysis of variance with the two within-sub-jects factors “data set” (1 vs. 2) and “presence ofeffect” (present vs. absent) yielded no significanteffect of the factor “presence of effect”, F(1, 11)¼ 2.59, MSE ¼ 0.208, p ¼ .14. Increasing thestatistical power of the experiment mightconfirm the expected difference (see Experiment2). Nevertheless, participants’ inferences exhibit asurprising grasp of the inferences that can bedrawn from the individual patterns of events.

A rather awkward pattern resulted in the “pre-diction before effect” condition. Contrary to ourpredictions participants did not predict the unob-served cause to be present more often, when theprobability of the effect in the observable cause’sabsence rose. They rather predicted the unob-served cause to be present less often in the obser-vables cause’s presence when the causal impact ofthe observable cause increased. In contrast to the“prediction after effect” condition it seems thatparticipants had no clue how to derive predictionsabout the unobserved cause in this condition.

Overall the results from Experiment 1 contra-dicted some of the theoretical assumptions madeby the currently predominant theories of causallearning. Participants assumed neither that theunobserved cause was always present nor that thetwo causes were independent (at least in theonline judgements). This finding challenges

associative theories, Power PC theory, and othercausal Bayes net theories, which assume indepen-dence of causes. However, we already havepointed out that an independence assumption isnot necessary for Power PC theory and causalBayes net theories in general. Causal Bayes nettheories including Power PC theory can modeldependence between observed and unobservedcauses. Therefore, even if participants’ answersdid not conform to the independence assumption,their answers still might be coherent with modi-fied versions of these theories. To be coherent par-ticipants’ estimates would have to honour theconstraints imposed by the learning data. Themost important constraint is that the product ofthe causal power (or causal strength) of the unob-served cause and the probability of the unobservedcause in the absence of the observed cause mustequal the probability of the effect in the absenceof the observed cause.

To find out whether participants honour thisconstraint we used their causal strength and theirfinal summary ratings of the unobserved cause torecalculate the probability of the effect E whencause C was absent:

Prec(ej�c)

¼ causal strength ratingA�rating P(aj�c):

The recalculated probabilities and the actuallyobserved probabilities are shown in Table 5. It canbe seen that the recalculated probabilities in the“prediction after effect” condition were surprisinglyclose to the actually observed probabilities. In con-trast, the recalculated probabilities in the other twoconditions were inaccurate. Since these ratings werecollected at the end of the learning phase thenecessary information to make accurate summaryestimates was available to learners in all conditions.However, it seems that participants had to be sensi-tized by the learning procedure to the presence andcausal strength of the unobserved cause to be able toderive coherent estimates. The procedure in the“prediction after effect” condition required partici-pants in each trial to think about the unobservedcause, and it provided participants with the right



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information to be able to do so. Without this infor-mation participants’ guesses showed some systema-ticity but they did not conform very well to theobserved data. Thus, simply asking participants tothink about unobserved causes on each trial—aswe did in the “prediction before effect” con-dition—was clearly not sufficient.

An interesting novel finding is the dissociationbetween online judgements and final summaryjudgements. Although the online judgementsshow deviations from the independence assump-tion, the final judgements are consistent withthis assumption. Apparently, online judgementswere driven by other aspects of the data than thesummary estimates. We discuss this point moreclosely in the General Discussion. This findingsupports normative theories of power estimation(e.g., Power PC theory) that hypothesize thatpeople default to an independence assumption.These theories might argue that the online judge-ments are less relevant than the final estimates, aspower estimates are typically not made online butrather on the basis of a learning sample. If peopleretrospectively make the independence assumptionwhen assessing power, their estimates should cor-respond to the predictions of normative theories(i.e., Power PC theory). This point is reinforcedby the fact that learners’ estimates seemed tomirror causal power rather than contingencies.

EXPERIMENT 2

In Experiment 1 participants could only passivelyobserve the occurrence of a cause and an effect.

Mere observations do not rule out the possibilitythat the observed cause is in fact related to asecond unobserved, confounding cause. It mighthave been the case that flowers were more ofteninfected by both microbes than by only one. Suchdependence might have even been plausible forsome participants. To make independence moresalient, we switched from observations to interven-tions in Experiment 2. In this experiment weallowed participants to arbitrarily manipulate theobservable cause. Since these random interventionscannot be based on the presence or absence of theunobserved cause, they are more likely to be inde-pendent from the unobserved cause (see Hagmayeret al., in press; Meder, Hagmayer, & Waldmann,2005; Waldmann & Hagmayer, 2005). Therefore,learning from interventions should make indepen-dence between alternative causes more salient thanshould learning from observations. We speculatedthat learners may be more prone to assume indepen-dence between alternative causes if they are allowedto freely manipulate the observable cause.

Participants in this second experiment wereinstructed to imagine being a captain on a pirateship firing his battery at a fortress. Their taskwas to assess their own hit rate—that is, thecausal strength of their actions upon the fortress.A second ship, which also fired at the fortressbut was occluded from participants’ view, servedas the unobserved cause. Participants had todecide whether to fire or not on each trial. Onlya limited number of shells were provided toensure that all participants received equivalentdata despite the fact that they arbitrarily set thecause themselves.

Table 5. Recalculated conditional probabilities of the effect in the absence of the observable cause


Condition Recalculated Observed Recalculated Observed Recalculated Observed

“Before effect” .13 (.13) .25 (.19) .24 (.17)

“After effect” .14 (.15) .10 .26 (.26) .30 .43 (.28) .50

Control .14 (.10) .21 (.20) .21 (.14)

Note: Calculations are based on the probabilities and causal strength parameters estimated by participants. Power PC theory is used to

integrate these estimates. See text for more details. Standard deviations in parentheses.



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The same two factors as those in the first exper-iment were manipulated. Learning conditionswere again varied between subjects. In the twoexperimental conditions participants either hadto guess whether the other ship had fired duringthe current trial before they were informed aboutthe occurrence of an explosion in the fortress(“prediction before effect”), or they had topredict the other ship’s action after they hadlearned whether the fortress was hit (“predictionafter effect”). In a third control condition no pre-dictions were requested. At the end of the learningphase all participants were requested to give expli-cit estimates of the probability of the unobservedcause conditional upon the manipulated cause.They also had to rate the causal strength of theobserved and of the unobserved cause.

As a second factor the data sets presented toparticipants were varied as a within-subjects vari-able. The order of the data sets was counterba-lanced as in the first experiment. Again theconditional probability of the effect in the presenceand absence of the cause was raised across con-ditions. In contrast to Experiment 1, in which con-tingencies were kept constant, we now kept causalpower constant. Therefore we expected partici-pants to rate the causal strength of the manipu-lated cause to be the same in all conditions.However, we again expected that the estimatedprobability and causal strength of the unobservedcause would rise in parallel to the probability ofthe effect in the absence of the observable cause.

Method

Participants and designA total of 60 students from the University ofGottingen, Germany, participated and were ran-domly assigned to one of the three learningconditions.

Materials and procedureThis second experiment was run on a computer.First, participants were instructed to imaginebeing a captain on a pirate ship trying to invadefortresses in the Caribbean. Therefore they werefiring at a fortress with their ship’s battery. A

second friendly ship, not visible from the captain’scurrent position, allegedly also fired at the fortress.Participants were told that they had the chance tofire 30 salvos at the fortress on 60 occasions. Theirtask was to assess how often their own batterywould hit the fortress. On each trial participantswere first asked whether they wanted to fire asalvo until they used up their shells. In the “predic-tion before effect” condition they were next askedto predict whether the other ship had fired on thisoccasion or not. No feedback was provided. Thenlearners observed whether a causal effect (a blastwithin the fortress) had occurred or not. In the“prediction after effect” condition participantswere informed after their intervention whetheran explosion at the fortress had occurred or not,and then had to predict whether the other shiphad also fired on this occasion. Again no feedbackwas given. In the control condition participantsonly decided whether to fire and then observedthe causal effect.

After completing the 60 trials participants wereasked to rate how often their salvos would hit thefortress if the other ship had stopped firing.Participants gave their answer on a scale rangingfrom 0 (¼ “never”) to 100 (¼ “always”). Theyalso had to estimate how often the other ship’sbattery would hit the fortress if they had stoppedfiring themselves. The same rating scale was usedagain. Participants were also asked how many ofthe 30 times they had fired had the other shipfired as well and how often within the set of 30trials in which they had not fired had the othership fired instead. No feedback was provided.Participants were then told that they had success-fully captured the fortress and sailed on to the next.Before the next data set was shown, it was empha-sized that the environmental conditions at the newfortress were completely different, so that othersuccess rates might result.

Three new data sets consisting of 60 cases eachwere constructed. Each was constructed in a waythat the contingency between the observed causeand the effect decreased across the data sets,whereas the causal power remained constant.Table 6 shows the conditional probabilities thatwere used to generate the data for each participant



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in the three data set conditions. Each set was gen-erated individually for each participant. Thismeans that the data sets were not fixed inadvance like in Experiment 1 but that the presenceof the effect was determined on each trial anewusing the probabilities shown in Table 6. Forexample, if a participant in the Data Set 1 con-dition chose to fire, the computer reported acausal effect with a probability of P ¼ .7. As inExperiment 1 the order of the data sets was coun-terbalanced across participants.

Results

First participants’ estimates of causal strength ofthe manipulated and the unobserved cause wereanalysed. Mean ratings for all conditions areshown in Table 7. An analysis of variance of thecausal strength ratings of the manipulated causewith learning condition as a between-subjectsfactor and data sets as a within-subjects factoryielded no significant effects. This result is inaccordance with the predictions of Power PCtheory because causal power was kept constantacross data sets. The same analysis conducted forthe causal strength estimates concerning the unob-served cause yielded two significant main effects.

As in Experiment 1 the estimated causal influencerose across the three data sets, F(2, 114) ¼ 65.7,MSE ¼ 408.2, p , .01. This finding is consistentwith all theories considered in the Introduction.There was also a significant difference betweenlearning conditions, F(2, 57) ¼ 4.06, MSE ¼591.8, p , .05. Participants in the “predictionafter effect” condition rated the causal strengthof the unobserved cause to be higher than that inthe other two conditions. This result may be a con-sequence of the learning procedure in thiscondition, which drew participants’ attention tothe possible influence of the unobserved cause.Remember that participants in this condition hadto predict the other cause’s presence based ontheir own action and the occurrence of the effect.Therefore every time the effect had occurredwithout the participant’s intervention, the partici-pant should have concluded that the other causemust have been present. No such predictions wererequired in the other two conditions, which mightbe the reason why participants in these conditionsmay have overlooked at least some of the crucialcases in which the effect had occurred on its own.

In a second step participants’ implicit assump-tions about the dependence between the manipu-lated and the unobserved cause were analysed.Thus again, the conditional frequencies generatedby participants during their trial-by-trial predic-tions were translated into conditional probabilitiesand subtracted to yield contingencies. The result-ing subjective dependencies are listed in Table 8.Supporting the results of Experiment 1, partici-pants tended to generate a negative dependencebetween the causes. With one exception (DataSet 1 in the “prediction after effect” condition) all

Table 7. Results of Experiment 2: Mean ratings of causal influence of observed and unobserved causes

Causal strength: Observed cause Causal strength: Unobserved cause


“Before effect” 67.7 (22.7) 65.1 (20.9) 68.0 (18.6) 27.3 (23.3) 44.8 (20.6) 65.8 (20.3)

“After effect” 70.0 (29.1) 61.4 (17.2) 62.0 (32.5) 36.0 (26.6) 57.9 (17.0) 81.6 (13.1)

Control 66.9 (15.8) 63.6 (11.8) 71.2 (17.6) 33.9 (32.7) 50.6 (21.2) 76.3 (16.8)


Table 6. Conditional probabilities used to generate the three data

sets of Experiment 2


P(ejc) .70 .80 .90

P(ej�c) .00 .33 .67

DP .70 .47 .23

Power pc .70 .70 .70



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generated dependencies deviated significantly fromzero. An analysis of variance with data set andlearning conditions as factors yielded a significantmain effect of data set, F(2, 76) ¼ 6.97, MSE ¼510.1, p , .01 and a significant interaction, F(2,76) ¼ 3.57, MSE ¼ 510.1, p , .05. The depen-dence became more negative when participantshad received effect information before their predic-tions but remained at the same negative level whenthey made their predictions before being informedabout the effect. In contrast to the generateddependencies in the online judgements the esti-mated dependencies in the final summary judge-ments did not statistically differ from each otherand from zero. Thus, as in the first experiment,there was a clear dissociation between online andfinal summary judgements. Whereas the onlinejudgements showed deviations from the indepen-dence assumption, the final estimates corre-sponded to it.

In a third analytical step we again analysed thetrial-by-trial online predictions more closely. As inExperiment 1, the generated patterns were trans-formed into subjective probabilities of the unob-served cause conditional upon the manipulatedcause, Pgen(AjC), in the “prediction before effect”condition and probabilities conditional upon pat-terns of the observed cause and the effect,Pgen(AjC,E), in the “prediction after effect” con-dition. The results are shown in Table 9. As inExperiment 1 it is important to note that DataSet 1 contained no cases in which the effectoccurred in the absence of the manipulated cause.Overall the results were similar to those inExperiment 1. The probabilities generated in the“prediction after effect” condition again conformed

to the implications of a Bayesian analysis.Participants predicted the unobserved cause to bepresent with a very high probability when theobservable cause was absent in the effect’s presenceand predicted it with a rather low probability whenboth cause and effect were absent. An analysis ofvariance with the two within-subject factors “dataset” (2 vs. 3) and “presence of effect” (present vs.absent) yielded a strong main effect of the presenceof the effect, F(1, 19) ¼ 115.4, MSE ¼ 0.071, p ,

.01, while all other effects were insignificant.Participants also differentiated between the casesin which both cause and effect were present andthe cases in which the cause was present withoutgenerating the effect. In accordance with theBayesian predictions participants inferred theunobserved cause with a higher probability in thefirst than in the second case. An analysis of variancewith the two within-subjects factors “data set” and“presence of effect” confirmed the significance ofthis difference, F(1, 19) ¼ 7.50, MSE ¼ 0.111, p, .05. The probabilities generated by participantsin the “prediction before effect” condition show auniform pattern. Learners predicted the unobservedcause to be present more often when the observedcause was absent than when it was present. Therewas apparently no difference between the threedata sets. Recall that in this condition participantshad to make their prediction before receiving infor-mation about the effect, which did not allow themto draw specific inferences for individual cases.Nevertheless, participants should have adaptedtheir predictions to the observable probability ofthe effect in the observable cause’s presence,which defines a lower boundary for the probabilityof the unobserved cause. This conditional

Table 8. Results of Experiment 2: Mean generated and estimated dependencies of observed and unobserved causes

Generated dependence Estimated dependence


“Before effect” –.24 (.319) –.33 (.380) –.29 (.438) –.04 (.203) .08 (.256) –.08 (.306)

“After effect” –.03 (.300) –.20 (.276) –.35 (.258) –.13 (.341) –.11 (.304) –.01 (.589)

Control — — — .10 (.366) .09 (.314) –.06 (.339)

Note: Values represent contingencies. Standard deviations in parentheses.



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Table 9. Results of Experiment 2: Mean probabilities of unobserved cause generated by participants conditional upon each pattern of observations

“Before effect” condition


Pgen(ajc) Pgen(aj�c) Pgen(ajc) Pgen(aj�c) Pgen(ajc) Pgen(aj�c)

.37 (.24) .61 (.26) .36 (.24) .69 (.24) .40 (.24) .69 (.26)

“After effect” condition


Pgen(aje.c) Pgen(aj�e.c) Pgen(aje.�c) Pgen(aj�e.�c) Pgen(aje.c) Pgen(aj�e.c) Pgen(aje.�c) Pgen(aj�e.�c) Pgen(aje.c) Pgen(aj�e.c) Pgen(aje.�c) Pgen(aj�e.�c)

.39 (.22) .29 (.21) — .40 (.29) .42 (.21) .26 (.30) .99 (.04) .39 (.32) .41 (.24) .18 (.25) .98 (.05) .29 (.29)

Note: Results concerning the “before effect” condition are shown in the upper half of the table, results concerning the “after effect” condition are shown in the lower half. Standard

deviations in parentheses.

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probability increased across trials. Therefore weexpected to find a difference between the data setsat least for trials in which the observable causewas absent. As in Experiment 1 participant’s pre-dictions showed no sensitivity to this implication.

Finally, we again investigated whether partici-pants’ final ratings honoured the constraintsimposed by causal Bayes net theories and PowerPC theory. The estimated probabilities andcausal strength estimates were used to recalculatethe probability of the effect in the absence of theobservable cause. If participants were sensitive tothe constraints imposed by these theories, thevalues should closely resemble the observed con-ditional probabilities. The results are shown inTable 10. It can be seen that participants tendedto honour the constraints. As in Experiment 1the best performance was found in the “predictionafter effect” condition in which participants (a)were asked about the unobserved cause on eachoccasion, and (b) had the information availablethat allowed them to make informed guesses.

GENERAL DISCUSSION

All current theories of causal reasoning considerunknown causes. However, there is no agreementabout the correct way to model inferences aboutsuch causes. The aim of this paper was to empiri-cally investigate the assumptions that learnersmake about unobserved causes to provide con-straints for theories of causal learning. In twoexperiments we requested participants to learn asingle causal relation between an observable

cause and an effect that, according to our instruc-tions, could also be caused by an unobserved cause.Participants never received any feedback about thisunobserved cause. Nevertheless, we asked them toassess the relation between the observed andthe unobserved cause and the causal strength ofboth the observed and the unobserved causeupon the effect.

Associative theories predict that the estimatedcausal influence of the observed cause should cor-respond to the observed contingency between theobserved cause and the effect. Power PC theoryand other causal Bayes net accounts, on the otherhand, predict that the estimated causal influenceof a cause should correspond to its causalpower—that is, its capacity to produce the effectin the absence of all other potential causes. Theresults of both experiments supported Power PCtheory over an associative account. Participants’estimates conformed to the pattern predicted bycausal power (see also Buehner et al., 2003).

All theoretical accounts agree that the causalstrength of the unobserved cause has to be atleast as high as the probability of the effect inthe observable cause’s absence, P(ej�c). However,whereas associative theories would predict thatthe estimated causal influence equals this prob-ability, Power PC theory and other causal Bayesnet accounts predict that this probability justmarks the lower boundary of the admissiblevalues of causal strength. We found evidence inboth experiments that the estimates of causalstrength of the unobserved cause increased pro-portional to P(ej�c). This finding replicates pre-vious results (Luhmann & Ahn, 2003) and is in

Table 10. Recalculated conditional probabilities of the effect in the absence of the observable cause


Condition Recalculated Observed Recalculated Observed Recalculated Observed

“Before effect” .13 (.13) .20 (.12) .42 (.22)

“After effect” .17 (.19) .00 .35 (.17) .33 .55 (.21) .67

Control .10 (.15) .24 (.13) .55 (.18)

Note: Calculations are based on the probabilities and causal strength parameters estimated by participants. Power PC theory is used to

integrate these estimates. Standard deviations in parentheses.



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accordance with all theories that were discussed inthe Introduction. However, we also found that theestimates were consistently higher than P(ej�c).This result favours causal Bayes net theories overan associative account.

Associative theories assume that an unobservedcause is permanently present. This assumptionensures that the unobserved cause is independentof the observed cause, which allows the causalstrength of both causes to be estimated. CausalBayes net theories including Power PC theory aremore flexible and allow the analysis of situationsin which the unobserved cause can be present orabsent. If independence is assumed, the causalpower of observable causes can be precisely deter-mined even when no information about the unob-served cause is available (see the power formula).In this case the probability of the unobservedcause and its causal strength are both constrainedby the observable conditional probabilities,especially the probability of the effect in the obser-vable cause’s absence, P(ej�c), which marks thelower boundary. In none of the experiments wasevidence found that participants assumed that theunobserved cause was permanently present.

However, the results also did not unanimouslysupport the independence assumption usuallymade by causal Bayes net accounts. In both experi-ments we found an interesting dissociation betweenparticipants’ online and final summary judgementsof the causes’ interdependence. While their finalestimates about the relation between the observedand the unobserved cause were consistent with anassumption of independence, their predictions forindividual trials were not. In Experiment 1 (obser-vational learning) participants generated a positivedependence if the effect only rarely occurred on itsown, but a negative dependence if it occurredfairly often without any apparent cause. InExperiment 2 (interventional learning) participantsgenerated a negative dependency throughout. Thisimplies that participants thought that the unob-served cause had occurred more often when theobservable cause was absent than when it waspresent. This finding is remarkable because inter-ventions should ensure that the manipulated causeoccurs independently of all unobserved causes.

There are several possible explanations for thisunexpected finding of a negative dependency inonline judgements. One reviewer speculated thatthe observed dependency might be due to thecharacteristics of the task. Predictions for individ-ual trials require translating subjective probabilitiesinto binary judgements. Such a translation processmight generate a bias towards dependency. Whilewe cannot rule out this possibility it seems unlikelyto us. Our analyses are based on findings of anextensive literature that people tend to matchprobabilities when generating sequences of events(see Hernstein, 1997; Myers, 1976; Vulkan,2000, for overviews). This assumption is supportedby the fact that the predictions that were based onobserved patterns of cause and effect (“predictionafter effect” condition) closely mirrored the predic-tions derived from a Bayesian analysis. There wasno sign of a bias in these judgements.

It is important to recall that the results concern-ing the online predictions clearly differed betweenthe two learning conditions. The predictions madebefore information about the presence of the effectwas revealed (“prediction before effect” condition)hardly showed any systematic relation to the datasets. The tendency to predict the unobservedcause less often when the observable cause wasalready present may be grounded in people’s reluc-tance to accept overdetermination of an effect.Since one cause suffices to explain an effect,assuming a second unobserved cause is not necess-ary. A second related intuition is that eventswhich are not causally related rarely occur simul-taneously by chance. Thus, it seems unlikely thatthe observed and the unobserved cause co-occur.This would also explain an overall tendency togenerate a negative dependency in this condition.

The tendency to create an increasingly negativedependency in the “prediction after effect” con-dition showed that participants in this conditionwere sensitive to the statistical properties of thedata sets. In both experiments participantsalmost every time predicted the presence of theunobserved cause when the effect had occurredon its own. In addition, participants tended toassume its absence when neither the observablecause nor the effect had occurred. When the



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observed cause and the effect were present partici-pants ascribed a slightly higher probability to theunobserved cause than when the observed causehad occurred without the effect. However, theoverall probability assigned to the unobservedcause was medium to low in both cases andvaried only slightly across conditions. These find-ings for the individual patterns of observationsexplain why participants tended to generate anincreasingly negative dependency in the “predic-tion after effect” condition across data sets.Across the data sets the number of trials inwhich the effect occurred on its own increased.Since participants assumed that some cause mustbe responsible for the observed effect, the risingnumber of these cases led to more predictions ofthe unobserved cause across data sets—that is,the probability of the unobserved cause in theobserved cause’s absence, Pgen(aj�c), rose substan-tially. In addition, the number of trials in whichcause and effect co-occurred also increased acrossdata sets. However, this trend only slightlyincreased the frequency of predictions of the unob-served cause in the observed cause’s presencebecause participants ascribed only a slightlyhigher probability to the unobserved cause in thepresence of both cause and effect than in trialswhen the cause was present by itself (see above).Therefore Pgen(ajc) rose only slightly. As a conse-quence a negative trend was observed across datasets. Thus, the finding of an increasingly negativedependency might be just a sign of participants’Bayesian reasoning about individual trials thatneglects the statistical properties of the wholesequence of events.

While participants’ online judgements deviatedsignificantly from independence, their finalsummary estimates did not. This dissociation pro-vides an interesting challenge for theories oncausal learning (see also Catena et al., 1998, for asimilar dissociation regarding causal strength esti-mates). As we have outlined in the previous para-graphs the dependence generated in the onlinejudgements is probably due to the way the predic-tions were derived for each individual case.Summary estimates, however, most likely do notfocus on individual cases but on larger samples of

cases. More research is needed to explore the pro-cesses underlying this interesting dissociation.

What are the implications of these findings forcausal Bayes net theories and Power PC theory?Since these theories model strength estimatesobtained at the asymptote of learning, summaryjudgements at the end of learning may be themore valid indicator of participants’ assumptionsabout unobserved causes at this point of learning.These estimates did not on average deviate fromindependence, which is consistent with PowerPC theory.

Even if the online judgements were viewed asthe more valid indicator of people’s intuitiveassumption about dependence, causal Bayes nettheories or Power PC theory are not refuted.Both theories may drop this assumption,however, at the cost of making causal power nolonger precisely estimable. Causal Bayes net the-ories can model cases in which unobserved andobserved causes are dependent. In this case, thesetheories provide constraints for consistent esti-mates. We used participants’ final estimates tofind out whether the estimates honour theseconstraints. The most important constraint isthat the probability of the effect in the absenceof the unobserved cause equals the product ofthe probability of the unobserved cause in theabsence of the observed cause and the causalpower of the unobserved cause, P(ej�c) ¼P(aj�c)pa (see Equations 2 and 5). Participants’estimates in fact tended to honour this constraint.Thus, even if independence was not assumed bylearners, their estimates were coherent with arational Bayesian analysis.

What are the implications of these findings forthe debate between associative and cognitive the-ories of causal learning? Our findings clearly chal-lenge associative theories (see Shanks, 2007)because participants proved capable of drawingsystematic inferences about the presence andcausal strength of cues that remained unobservablethroughout learning. This finding supports theassumption of rational theories of causal learning(e.g., Power PC theory) that people representcausal learning tasks as situations in which unob-servable causes and observed causes jointly



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generate the observed effects within a common-effect model. By contrast, associative theoriesclaim that observed causes compete with the con-stantly present and observable context withoutseparately considering unobservable causal events.Our results add to other findings reported in thisvolume, which show that participants usethe observable information to draw rational infer-ences about observable and unobservable causalevents and are sensitive to the structure of causalmodels (see Booth & Buehner, 2007; Cobos,Lopez, & Luque, 2007; De Houwer, Vandorpe,& Beckers, 2007; Vandorpe, De Houwer, &Beckers, 2007).

To sum up, our results contradict the assump-tion of associative theories that learners assumethe constant presence of alternative, unobservedcauses. The results about the independenceassumption made by Bayesian theories are mixed.Whereas the online estimates violated indepen-dence, occasionally in the direction of a positive,more often in the direction of a negative corre-lation, the summary estimates were on averageclose to independence. Moreover, a Bayesiananalysis revealed that the estimates were rationaland were consistent with Bayesian constraints.These analyses provide further convincing evi-dence for the usefulness of a Bayesian analysis ofcausal learning.

REFERENCES

Ahn, W.-K., Marsh, J. K., & Luhmann, C. C. (in press).Dynamic interpretations of covariation data. In A.Gopnik & L. Schulz (Eds.), Causal learning:

Psychology, philosophy, and computation. Oxford, UK:Oxford University Press.

Audi, R. (1995). The Cambridge dictionary of

philosophy. Cambridge, MA: Cambridge UniversityPress.

Booth, S. L., & Buehner, M. J. (2007). Asymmetries incue competition in forward and backward blockingdesigns: Further evidence for causal model theory.Quarterly Journal of Experimental Psychology, 60,387–399.

Buehner, M. J., & Cheng, P. W. (1997). Causalinduction: The power PC theory versus the

Rescorla–Wagner model. In P. Langley & M. G.Shafto (Eds.), Proceedings of the Nineteenth Annual

Conference of the Cognitive Science Society (pp. 55–60). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Buehner, M. J., Cheng, P. W., & Clifford, D. (2003).From covariaton to causation: A test of the assump-tion of causal power. Journal of Experimental

Psychology: Learning, Memory, and Cognition, 29,1119–1140.

Catena, A., Maldonado, A., & Candido, A. (1998). Theeffect of frequency of judgement and the type of trialson covariation learning. Journal of Experimental

Psychology: Human Perception & Performance, 24,481–495.

Cheng, P. W. (1997). From covariation to causation:A causal power theory. Psychological Review, 104,367–405.

Cobos, P. L., Lopez, F. J., & Luque, D. (2007).Interference between cues of the same outcomedepends on the causal interpretation of the events.Quarterly Journal of Experimental Psychology, 60,369–386.

Danks, D. (2003). Equilibria of the Rescorla–Wagnermodel. Journal of Mathematical Psychology, 47,109–121.

Danks, D. (in press). Causal learning from observationsand manipulations. In M. Lovett & P. Shah (Eds.),Thinking with data. Mahwah, NJ: LawrenceErlbaum Associates, Inc.

De Houwer, J., Vandorpe, S., & Beckers, T. (2007).Statistical contingency has a different impact onpreparation judgements than on causal judgements.Quarterly Journal of Experimental Psychology, 60,418–432.

Gelman, S. (2003). The essential child. Origins of essenti-

alism in everyday thought. Cambridge, MA: OxfordUniversity Press.

Glymour, C. (2001). The mind’s arrow. Bayes nets and

graphical causal models in psychology. Cambridge,MA: MIT Press.

Gopnik, A., Glymour, C., Sobel, D. M., Schulz, L. E.,Kushnir, T., & Danks, D. (2004). A theory of causallearning in children: Causal maps and Bayes nets.Psychological Review, 111, 3–32.

Gopnik, A., & Meltzoff, A. N. (1997). Words, thoughts

and theories. Cambridge, MA: MIT Press.Gopnik, A., & Wellman, H. M. (1994). The

theory theory. In L. Hirschfeld & S. Gelman(Eds.), Mapping the mind: Domain specificity in

cognition and culture (pp. 257–293). New York:Cambridge University Press.



Dow

nloa

ded

By: [

Sub

- BBM

] At:

16:2

6 26

Apr

il 20

07

Griffiths, T. L., & Tenenbaum, J. B. (2005). Structureand strength in causal induction. Cognitive

Psychology, 51, 285–386.Hagmayer, Y., Sloman, S. A., Lagnado, D. A., &

Waldmann, M. R. (in press). Causal reasoningthrough intervention. In A. Gopnik & L. Schulz(Eds.), Causal learning: Psychology, philosophy, and

computation. Oxford, UK: Oxford University Press.Hernstein, R. J. (1997). The matching law. Cambridge,

MA: Harvard University Press.Jensen, F. V. (1997). An introduction to Bayesian net-

works. London: Springer.Keinath, A., & Krems, J. F. (1999). Anomalous data

integration in diagnostic reasoning. In A. Bagnara(Ed.), Proceedings of the 1999 European Conference

on Cognitive Science (pp. 213–218). Siena, Italy:Consiglo Nationale delle Ricerche.

Krems, J. F., & Johnson, T. (1995). Integration ofanomalous data in multi causal explanations. InJ. D. Moore & J. F. Lehman (Eds.), Proceedings of

the Seventeenth Annual Conference of the Cognitive

Science Society (pp. 277–282). Hillsdale, NJ:Lawrence Erlbaum Associates, Inc.

Kushnir, T., Gopnik, A., Schulz, L., & Danks, D.(2003). Inferring hidden causes. In P. Langley &M. G. Shafto (Eds.), Proceedings of the Twenty-fifth

Annual Conference of the Cognitive Science Society.

Mahwah, NJ: Lawrence Erlbaum Associates, Inc.Lober, K., & Shanks, D. R. (2000). Is causal induction

based on causal power? Critique of Cheng (1997).Psychological Review, 107, 195–212.

Luhmann, C. C., & Ahn, W.-K. (2003). Evaluating thecausal role of unobserved variables. In P. Langley &M. G. Shafto (Eds.), Proceedings of the Twenty-fifth

Annual Conference of the Cognitive Science Society.

Mahwah, NJ: Lawrence Erlbaum Associates, Inc.Meder, B., Hagmayer, Y., & Waldmann, Y. (2005).

Doing after seeing. In B. G. Bara, L. Barsalou, &M. Bucciarelli (Eds.), Proceedings of the Twenty-

Seventh Annual Conference of the Cognitive Science

Society. Mahwah, NJ: Lawrence Erlbaum Associates,Inc.

Medin, D. L., & Ortony, A. (1989). Psychologicalessentialism. In S. Vosniadou & A. Ortony (Eds.),Similarity and analogical reasoning (pp. 179–195).Cambridge, UK: Cambridge University Press.

Myers, J. L. (1976). Probability learning and sequencelearning. In W. K. Estes (Ed.), Handbook of learning

and cognitive processes: Approaches to human learning

and motivation (pp. 171–205). Hillsdale, NJ:Lawrence Erlbaum Associates, Inc.

Pearl, J. (1988). Probabilistic reasoning in intelligent

systems: Networks of plausible inference. SanFrancisco, CA: Morgan Kaufmann.

Pearl, J. (2000). Causality: Models, reasoning, and

inference. Cambridge, MA: Cambridge UniversityPress.

Rehder, B. (2003). A causal-model theory of conceptualrepresentation and categorization. Journal of

Experimental Psychology: Learning, Memory, and

Cognition, 29, 1141–1159.Rehder, B., & Hastie, R. (2001). Causal knowledge and

categories: The effects of causal beliefs on categoriz-ation, induction, and similarity. Journal of

Experimental Psychology: General, 130, 323–360.Rescorla, R. A., & Wagner, A. R. (1972). A theory of

Pavlovian conditioning: Variations in the effective-ness of reinforcement and non-reinforcement. InA. H. Black & W. F. Prokasy (Eds.), Classical con-

ditioning II. Current research and theory (pp. 64–99). New York: Appleton-Century-Crofts.

Shanks, D. R. (1987). Acquisition functions in contin-gency judgment. Learning and Motivation, 18,147–166.

Shanks, D. R. (2007). Associationism and cognition:Human contingency learning at 25. Quarterly

Journal of Experimental Psychology, 60, 291–309.Shanks, D. R., Holyoak, K. J., & Medin, D. L.

(1996). The psychology of learning and motivation:

Vol. 34. Causal learning. San Diego, CA: AcademicPress.

Sloman, S. A., & Lagnado, D. A. (2005). Do we “do”?Cognitive Science, 29, 5–39.

Spirtes, P., Glymour, C., & Scheines, R. (1993).Causation, prediction and search. New York: Springer.

Tenenbaum, J. B., & Griffiths, T. L. (2003). Theory-based causal inference. In T. K. Leen, T. G.Dietterich, & V. Tresp (Eds.), Advances in neural

information processing systems (Vol. 15, pp. 35–42).Cambridge, MA: MIT Press.

Vandorpe, S., De Houwer, J., & Beckers, T. (2007).Outcome maximality and additivity trainingalso influence cue competition in causal learningwhen learning involves many cues and events.Quarterly Journal of Experimental Psychology, 60,356–368.

Vulkan, N. (2000). An economist’s perspective onprobability matching. Journal of Economic Surveys,14, 101–118.

Waldmann, M. R., & Hagmayer, Y. (2001). Estimatingcausal strength: The role of structural knowledge andprocessing effort. Cognition, 82, 27–58.



Dow

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Waldmann, M. R., & Hagmayer, Y. (2005). Seeing vs.doing: Two modes of accessing causal knowledge.Journal of Experimental Psychology: Learning,

Memory, and Cognition, 31, 216–227.Wasserman, E. A., Elek, S. M., Chatlosh, D. L., &

Baker, A. G. (1993). Rating causal relations: Roleof probability in judgments of response–outcomecontingency. Journal of Experimental Psychology:

Learning, Memory, and Cognition, 19, 174–188.

Wellman, H. M. (1990). The child’s theory of mind.Cambridge, MA: MIT Press.

White, P. A. (2003). Making causal judgments from theproportion of confirming instances: The pCI rule.Journal of Experimental Psychology: Learning,

Memory, and Cognition, 29, 710–727.Woodward, J. (2003). Making things happen. A theory of

causal explanation. Oxford, UK: Oxford UniversityPress.



the quarterly journal of experimental psychology...observed cause—that is, dp ¼ p(ejc)–p(ej c);...

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