Structure learning in human causal induction

Transcription

1 Structure lerning in humn cusl induction Joshu B. Tenenbum & Thoms L. Griffiths Deprtment of Psychology Stnford University, Stnford, CA jbt,gruffydd psych.stnford.edu Abstrct We use grphicl models to explore the question of how people lern simple cusl reltionships from dt. The two leding psychologicl theories cn both be seen s estimting the prmeters of fixed grph. We rgue tht complete ccount of cusl induction should lso consider how people lern the underlying cusl grph structure, nd we propose to model this inductive process s Byesin inference. Our rgument is supported through the discussion of three dt sets. 1 Introduction Cuslity plys centrl role in humn mentl life. Our behvior depends upon our understnding of the cusl structure of our environment, nd we re remrkbly good t inferring custion from mere observtion. Constructing forml models of cusl induction is currently mjor focus of ttention in computer science [7], psychology [3,6], nd philosophy [5]. This pper ttempts to connect these litertures, by frming the debte between two mjor psychologicl theories in the computtionl lnguge of grphicl models. We show tht existing theories equte humn cusl induction with mximum likelihood prmeter estimtion on fixed grphicl structure, nd we rgue tht to fully ccount for humn behviorl dt, we must lso postulte tht people mke Byesin inferences bout the underlying cusl grph structure itself. Psychologicl models of cusl induction ddress the question of how people lern ssocitions between cuses nd effects, such s, the probbility tht some event cuses outcome. This question might seem trivil t first; why isn t simply, the conditionl probbility tht occurs ( s opposed to ) given tht occurs? But consider the following scenrios. Three cse studies hve been done to evlute the probbility tht certin chemicls, when injected into rts, cuse certin genes to be expressed. In cse 1, levels of gene 1 were mesured in 100 rts injected with chemicl 1, s well s in 100 uninjected rts; cses nd 3 were conducted likewise but with different chemicls nd genes. In cse 1, 40 out of 100 injected rts were found to hve expressed the gene, while 0 out of 100 uninjected rts expressed the gene. We will denote these results s "#"%#&. Cse produced the results ('#"#&) ", while cse 3 yielded +*,#"#&-/.#"(. For ech cse, we would like to know the probbility tht the chemicl cuses the gene to be expressed,, where denotes the chemicl nd denotes gene expression.

2 People typiclly rte 0 1 highest for cse 1, followed by cse nd then cse 354/ In n experiment described below, these cses received men rtings (on 0-0 scle) of, 8" ', nd, respectively. Clerly 1 ;: < 0 = 9, becuse cse 3 hs the highest vlue of 0 = 9 but receives the lowest rting for. The two leding psychologicl models of cusl induction elborte upon this bsis in ttempting to specify >. The?7 model [6] clims tht people estimte ccording to?7 < 0 AB 0 (We restrict our ttention here to fcilittory cuses, in which cse?7 is lwys between 0 nd 1.) Eqution 1 cptures the intuition tht is perceived to cuse to the extent tht s occurence increses the likelihood of observing. Recently, Cheng [3] hs identified severl shortcomings of?7 nd proposed tht 0 > insted corresponds to cusl power, the probbility tht produces in the bsence of ll other cuses. Formlly, the power model cn be expressed s: CED(FHGJI?7 There re vriety of normtive rguments in fvor of either of these models [3,7]. Empiriclly, however, neither model is fully dequte to explin humn cusl induction. We will present mple evidence for this clim below, but for now, the bsic problem cn be illustrted with the three scenrios bove. While people rte higher for cse, 7/100,0/100, thn for cse 3, 53/100,46/100,?7 rtes them eqully nd the power model rnks cse 3 over cse. To understnd this discrepncy, we hve to distinguish between two possible senses of 1 : the probbility tht C cuses E (on ny given tril when C is present) versus the probbility tht C is cuse of E (in generl, s opposed to being cuslly independent of E). Our clim is tht the?7 nd power models concern only the former sense, while people s intuitions bout > re often concerned with the ltter. In our exmple, while the effect of on ny given tril in cse 3 my be equl to (ccording to?7 ) or stronger thn (ccording to power) its effect in cse, the generl pttern of results seems more likely in cse thn in cse 3 to be due to genuine cusl influence, s opposed to spurious correltion between rndom smples of two independent vribles. In the following section, we formlize this distinction in terms of prmeter estimtion versus structure lerning on grphicl model. Section 3 then compres two vrints of our structure lerning model with the prmeter estimtion models (?7 nd power) in light of dt from three experiments on humn cusl induction. Grphicl models of cusl induction The lnguge of cusl grphicl models provides useful frmework for thinking bout people s cusl intuitions [5,7]. All the induction models we consider here cn be viewed s computtions on simple directed grph (K I)LC"MEN in Figure 1). The effect node is the child of two binry-vlued prent nodes:, the puttive cuse, nd O, constnt bckground. Let PQSRT N N)U % RV XW W U denote sequence of Y trils in which nd re ech observed to be present or bsent; O is ssumed to be present on ll trils. (To keep nottion concise in this section, we use 1 or 0 in ddition to or to denote presence or bsence of n event, e.g. [Z\ (1) () if the cuse is present on the ] th tril.) Ech prent node is ssocited with prmeter, ^X_ or ^`, tht defines the strength of its effect on. In the

3 d k m W W N?7 model, the probbility of occuring is liner function of : (b%^ _ ^ ` c^ _ed ^ `gf (We use to denote model probbilities nd for empiricl probbilities in the smple P.) In the cusl power model, s first shown by Glymour [5], is noisy-or gte: b)^x_ ^`99 h 1^X_=[.1 Prmeter inferences:?7 nd Cusl Power 1^`-i In this frmework, both the?7 nd power model s predictions for cn be seen s mximum likelihood estimtes of the cusl strength prmeter ^` in K I)L(C&MjN, but under different prmeteriztions. For either model, the loglikelihood of the dt is given by TP ^ _ ^ ` l D#q r Zon Njp m Zon N JZ D#q p Z Z --s[t Z %Z d Z Z - vjz D q p (3) (4) s%tou (5) Z %ZTw where we hve suppressed the dependence of Z Z on ^ _ ^ `. Breking this sum into four prts, k one for ech possible combintion of (x nd x tht could be observed, VPB ^ _ ^ ` cn be written s Yy 0 97z5 9 D#q p { 9 d v J}w D#q p { 9-V~ (7) Y }7z5 \ D#q p { \ d }- D q p { 9-V~ By the Informtion inequlity [4], Eqution 7 is mximized whenever ^ _ nd ^ ` cn be chosen to mke the model probbilities equl to the empiricl probbilites: b%^x_ b%^ _ ^`9 ^ ` To show tht the?7 model s predictions for 0 > correspond to mximum likelihood estimtes of ^ ` under liner prmeteriztion of K I%L(C"M N, we identify ^ ` in Eqution 3 with?7 (Eqution 1), nd ^ _ with { \. Eqution 3 then reduces to { 9 for the cse { (i.e., { ) nd to + for the cse (i.e., { ), thus stisfying the sufficient conditions in Equtions 8-9 for ^ _ nd ^ ` to be mximum likelihood estimtes. To show tht the cusl power model s predictions for correspond to mximum likelihood estimtes of ^ ` under noisy-or prmeteriztion, we follow the nlogous procedure: identify ^` in Eqution 4 with power (Eqution ), nd ^X_ with }. Then Eqution 4 reduces to 9 for J nd to + for c, gin stisfying the conditions for ^X_ nd ^` to be mximum likelihood estimtes.. Structurl inferences: Cusl Support nd 9ƒ The centrl clim of this pper is tht people s judgments of reflect something other thn estimtes of cusl strength prmeters the quntities tht we hve just shown to be computed by?7 nd the power model. Rther, people s judgments my correspond to inferences bout the underlying cusl structure, such s the probbility tht is direct (6) (8) (9)

4 cuse of. In terms of the grphicl model in Figure 1, humn cusl induction my be focused on trying to distinguish between K I%L(C"M N, in which is prent of, nd the null hypothesis of K I)L(C&M3, in which is not. This structurl inference cn be formlized s Byesin decision. Let ` be binry vrible indicting whether or not the link exists in the true cusl model responsible for generting our observtions. We will ssume noisy-or gte, nd thus our model is closely relted to cusl power. However, we propose to model humn estimtes of s cusl support, the log posterior odds in fvor of K I)L(C&MjN 3`g over K I)LC"M" 3`1 : - &C"C3D I-ˆ D q 0 E` p 0 E` Vi Byes rule, we cn express ` Pe in terms of the mrginl likelihood or evidence, VP `, nd the prior probbility tht is cuse of, ` : Pe Pe 3`g PeŠ< 0 TP 3`g - 3` For now, we tke E`Œ Ž 3` integrting the likelihood TP ^X_ N N 0 TP 3`g 9< TP ^X_ ^`93 ^X_ ^` 3`g (10) (11). Computing the evidence requires ^`9 over ll possible vlues of the strength prmeters: \ /^X_ /^` We tke j ^X_ ^` 3`g k to be uniform density, nd we note tht j VP ^X_ ^` is simply the exponentil of TP ^X_ ^`9 s defined in Eqution 5. VP E`1, the mrginl likelihood for K I)LC"M3, is computed similrly, but with the prior j ^X_ ^`= E`Ž in Eqution 1 replced by j ^X_ E` - ^`x. We gin tke j ^X_ 3` to be uniform. The Dirc delt distribution on ^` enforces the restriction tht the link is bsent. By mking these ssumptions, we eliminte the need for ny free numericl prmeters in our probbilistic model (in contrst to similr Byesin ccount proposed by Anderson [1]). Becuse cusl support depends on the full likelihood functions for both K I%L(C"M N nd K I%L(C"M3, we my expect the support model to be modulted by cusl power which is bsed strictly on the likelihood mximum estimte for K I%L(C"MEN but only in interction with other fctors tht determine how much of the posterior probbility mss for ^` in K I)L(C&M N is bounded wy from zero (where it is pinned in K I)LC"M3 ). In generl, evluting cusl support my require firly involved computtions, but in the limit of lrge Y (1) nd wek cusl strength ^`, it cn be pproximted by the fmilir }ƒ sttistic for independence, Y iw s- iw s- o o. Here iw sh š \ iw œ- ž - 0 is the fctorized pproximtion to sw, which /œ- ssumes nd to be independent (s they re in K I%L(C"ME ). 3 Comprison with experiments In this section we exmine the strengths nd weknesses of the two prmeter inference models,?7 nd cusl power, nd the two structurl inference models, cusl support nd ƒ, s ccounts of dt from three behviorl experiments, ech designed to ddress different spects of humn cusl induction. To compenste for possible nonlinerities in people s use of numericl rting scles on these tsks, ll model predictions hve been scled by power-lw trnsformtions, Ÿj 3 - šq# E9, with chosen seprtely for ech model

5 nd ech dt set to mximize their liner correltion. In the figures, predictions re expressed over the sme rnge s the dt, with minimum nd mximum vlues ligned. Figure presents dt from study by Buehner & Cheng [], designed to contrst the predictions of?7 nd cusl power. People judged 1 for hypotheticl medicl studies much like the gene expression scenrios described bove, seeing eight cses in which occurred nd eight in which did not occur. Some trends in the dt re clerly cptured by the 5#"%"5' cusl* power model but not by?7, such s the monotonic decrese in 1 from 5 #*&)& to, s?7 stys constnt but 0 { \ (nd hence power) decreses (columns 6-9). Other trends re clerly cptured by?7 but not by the power model, like the monotonic #5 increse "#5# in s 0 { 9 stys constnt t 1.0 but 0 = 9 decreses, from "%"5# to (columns 1, 6, 10, 13, 15). However, one of the most slient trends is cptured by neither model: the decrese in 1 s?7 stys constnt t 0 but } decreses (columns 1-5). The cusl support model predicts this decrese, s well s the other trends. The intuition behind the model s predictions for?7 c is tht decresing the bse rte \ increses the opportunity to observe the cuse s influence nd thus increses the sttisticl force behind the inference tht does not cuse, given?7. This effect is most obvious when 0 { 9{ { \{, yielding ceiling effect with no sttisticl leverge [3], but lso occurs to lesser extent for }X. While 9ƒ generlly pproximtes the support model rther well, it lso fils to explin the cses with { 99< +, which lwys yield }ƒ. The "54#* superior fit of the support model is reflected in its correltion with the dt, giving ;ƒ while the power,?7, nd }ƒ models gve ;ƒ vlues of & 8&, "58#, nd "58# respectively. Figure 3 shows results from n experiment conducted by Lober nd Shnks [6], designed to explore the trend in Buehner nd Cheng s experiment tht ws predicted by?7 but not by the power model. Columns 4-7 replicted the monotonic increse in when remins constnt t 1.0 but +} decreses, this time with 8 cses in which occurred nd 8 in which did not occur. Columns 1-3 show second sitution in which the predictions of the power model re constnt, but judgements of increse. Columns 8-10 feture three scenrios with equl?7, for which the cusl power model predicts decresing trend. These effects were explored by presenting totl of 60 trils, rther thn the 56 used in Columns 4-7. For ech of these trends the?7 model outperforms the cusl power model, with overll ƒ "54. "5,. vlues of nd respectively. However, it is importnt "%"5.# to note % (" tht 8#"w" the/responses % +& &-" of# the humn subjects in columns 8-10 (contingencies ) re not quite consistent with the predictions of "58?7 "%" : they show slight U-shped non-linerity, with 0 > judged to be smller for thn for either of the extreme cses. This trend is predicted by the cusl support model nd its }ƒ pproximtion, however, which both give the slightly better ;ƒ of Figure 4 shows dt tht we collected in similr survey, iming to explore this non-liner effect in greter depth. 35 students in n introductory psychology clss completed the survey for prtil course credit. They ech provided judgment of in 14 different medicl scenrios, where informtion bout 9 nd { \ ws provided in terms of how mny mice from smple of 100 expressed prticulr gene. Columns 1-3, 5-7, nd 9-11 show contingency structures designed to elicit U-shped trends in. Columns 4 nd 8 give intermedite vlues, lso consistent with the observed non-linerity. Column 14 ttempted to explore the effects of mnipulting smple size, with contingency structure of ('# '"%4#,#&4,. In ech cse, we observed the predicted nonlinerity: in set of situtions with the sme?7, the situtions involving less extreme probbilities show reduced judgments of >. These non-linerities re not consistent with the?7 model, but

6 re predicted"54# by both cusl support nd ƒ.?7 ctully chieves correltion comprble to 9ƒ ( ƒ for both models) becuse the non-liner effects contribute only&wekly 8 to the totl vrince. The support model gives slightly worse "5,#8 fit thn 9ƒ, ƒ, while the power model gives poor ccount of the dt, ;ƒ 4 Conclusions nd future directions In ech of the studies bove, the structurl inference models bsed on cusl support or }ƒ consistently outperformed the prmeter estimtion models,?7 nd cusl power. While cusl power nd?7 were ech cpble of cpturing certin trends in the dt, cusl support ws the only model cpble of predicting ll the trends. For the third dt set, }ƒ provided significntly better fit to the dt thn did cusl support. This finding merits future investigtion in study designed to tese prt }ƒ nd cusl support; in ny cse, due to the close reltionship between the two models, this result does not undermine our clim tht probbilistic structurl inferences re centrl to humn cusl induction. One unique dvntge of the Byesin cusl support model is its bility to drw inferences from very few observtions. We hve begun line of experiments, inspired by Gopnik, Sobel & Glymour (submitted), to exmine how dults revise their cusl judgments when given only one or two observtions, rther thn the lrge smples used in the bove studies. In one study, subjects were fced with mchine tht would inform them whether pencil plced upon it contined superled or ordinry led. Subjects were either given prior knowledge tht superled ws rre or tht it ws common. They were then given two pencils, nlogous to O nd in Figure 1, nd sked to rte how likely these pencils were to hve superled, tht is, to cuse the detector to ctivte. Men responses reflected the induced prior. Next, they were shown tht the superled detector responded when O nd were tested together, nd their cusl rtings of both O nd incresed. Finlly, they were shown tht O set off the superled detector on its own, nd cusl rtings of O incresed to ceiling while rtings of returned to their prior levels. This sitution is exctly nlogous to tht explored in the medicl tsks described bove, nd people were ble to perform ccurte cusl inductions given only one tril of ech type. Of the models we hve considered, only Byesin cusl support cn explin this behvior, by llowing the prior in Eqution 11 to dpt depending on whether superled is rre or common. We lso hope to look t inferences bout more complex cusl structures, including those with hidden vribles. With just single cuse, cusl support nd 9ƒ re highly correlted, but with more complex structures, the Byesin computtion of cusl support becomes incresingly intrctble while the }ƒ pproximtion becomes less ccurte. Through experiments with more complex structures, we hope to discover where nd how humn cusl induction strikes blnce between ponderous rtionlity nd efficient heuristic. Finlly, we should stress tht despite the superior performnce of the structurl inference models here, in mny situtions estimting cusl strength prmeters is likely to be just s importnt s inferring cusl structure. Our hope is tht by using grphicl models to relte nd extend upon existing ccounts of cusl induction, we hve provided frmework for exploring the interply between the different kinds of judgments tht people mke. References ª w«[«j. Anderson (1990). The dptive chrcter of thought. Erlbum. M. Buehner & P. Cheng (1997) Cusl induction; The power PC theory versus the Rescorl- Wgner theory. In Proceedings of the 19th Annul Conference of the Cognitive Science Society..

7 [«J«[«[«±[«P. Cheng (1997). From covrition to custion: A cusl power theory. Psychologicl Review 104, T. Cover & J.Thoms (1991). Elements of informtion theory. Wiley. C. Glymour (1998). Lerning cuses: Psychologicl explntions of cusl explntion. Minds nd Mchines 8, K. Lober & D. Shnks (000). Is cusl induction bsed on cusl power? Critique of Cheng (1997). Psychologicl Review 107, J. Perl (000). Cuslity. Cmbridge University Press. ²³² ²³² ²³² Grph 1 (h C = 1) B C w B w C ³ w B Grph 0 (h C = 0) ³ B C ³ P(e+ c+) P(e+ c ) Humns P E E Model Form of P(e b,c) P(C >E) P Liner w C Power Noisy OR gte w C Support Noisy OR gte log P(h C = 1) P(h C = 0) N/A Power Support χ Figure 1: Different theories of humn cusl induction expressed s different opertions on simple grphicl model. The P nd power models correspond to mximum likelihood prmeter estimtes on fixed grph (Grph 1 ), while the support model corresponds to (Byesin) inference bout which grph is the true cusl structure. Figure : Computtionl models compred with the performnce of humn prticipnts from Buehner nd Cheng [1], Experiment 1B. Numbers long the top of the figure show stimulus contingencies. P(e+ c+) P(e+ c ) Humns P P(e+ c+) P(e+ c ) Humns P Power Power N/A Support Support χ χ Figure 3: Computtionl models compred with the performnce of humn prticipnts from Lober nd Shnks [5], Experiments 4 6. Figure 4: Computtionl models compred with the performnce of humn prticipnts on set of stimuli designed to elicit the non monotonic trends shown in the dt of Lober nd Shnks [5].