Using Ideal Observers in Higher-order Human Category Learning

Transcription

1 Usig Ideal Observers i Higher-order Huma ategory Learig Aiket Kittur (kittur@ucla.edu) Keith J. Holyoak (holyoak@lifesci.ucla.edu) Departmet of Psychology, Uiversity of aliforia, Los Ageles A Joh E. Hummel (jehummel@cyrus.psych.uiuc.edu) Departmet of Psychology, Uiversity of Illiois at Urbaa-hampage, Urbaa IL 6820 Abstract Ideal observer models have prove useful i ivestigatig assumptios about huma iformatio processig i a variety of perceptual tasks. However, these models have ot bee applied i the area of higher-order category learig. We describe a simple Bayesia ideal observer ad apply it to empirical data o category learig. We describe a experimet i which we foud that acquisitio of family resemblace categories was drastically impaired if the categories were defied by relatios betwee features rather tha by the features themselves. A ideal observer was used to test whether this effect could be accouted for by iheret iformatio differeces betwee the coditios. A compariso of participats performace to the model foud a sigificat differece i efficiecy of learig eve after accoutig for iformatio differeces betwee coditios. This aalysis illustrates how ideal observer methods ca provide useful tools for aalyzig higher-order category learig. Keywords: categorizatio, category learig, relatios, features, ideal observer Itroductio A ideal observer model makes optimal use of a set of give iformatio i performig a task. Such models have traditioally prove useful i ivestigatig huma iformatio use i various perceptual tasks by providig a upper boud or bechmark by which to measure performace. If a huma ca perform at the same level as (or better tha) the ideal model, the we kow that the huma is makig use of all of the available iformatio i the situatio (or, i the case of humas outperformig the ideal, more iformatio tha was available to the model). If humas uderperform the ideal model, the differece ca ofte highlight specific costraits that limit huma iformatio processig. The degree to which huma performace approaches that of a ideal observer ca provide a measure of processig efficiecy. Ideal observers have most commoly bee applied to uderstadig huma low-level visual tasks ivolvig detectio ad discrimiatio (see Geisler, 2003), though they have also bee applied to tasks such as readig (Legge, Klitz, & Tja, 997), object recogitio (Liu, Kill, & Kerste, 995), ad reachig (Trommershäuser, Gepshtei, Maloey, Lady, & Baks, 2004). However, few studies have applied ideal observer methods to higher-order cogitive tasks, at least i part because of the difficulty of specifyig exactly what is ideal. Istead, most studies of huma category learig compare coditios agaist each other ad assume that differeces i performace capture theoretically-cetral differeces betwee coditios. However, there may be differeces betwee coditios that are ot relevat to the variable beig measured (e.g., oise). Ideal observers ca provide a theoretical upper boud o huma performace (give a set of assumptios), ad ca be used to cotrol for some of these extraeous variables. I this paper we describe a simple method for creatig a ideal observer model that takes as iput features, ad relatios betwee features, of the sort commoly used i category learig studies with artificial stimuli. The ideal observer assumes that the experimeter (but ot ecessarily the learer) has full kowledge of the geeratig model used to costruct stimuli based o these features ad relatios. This assumptio is typically met i experimetal category learig paradigms i which artificial stimuli are used. We will first describe the model ad the apply it by simulatig performace i a actual category learig experimet. The Model The model uses a Bayesia framework to assig stimuli to categories ad to lear from labeled feedback. We use a versio of a aïve Bayesia classifier, oe of the simplest probabilistic classifiers, which is optimal whe all iput features are idepedet (ad ca eve be optimal i certai less restricted circumstaces; see Domigos & Pazzai, 997). The aïve Bayesia classifier makes the assumptio that all features of a give category are geerated idepedetly, that is:, F,..., F ) = ) F )... F ) () for class variable (which represets all possible categories) ad feature variables through. Applyig Bayes rule results i the followig equatio: F,..., F ) = ) i= [ ) F ) i= i F )] The deomiator i Equatio 2 is a ormalizatio costat that is idetical for all categories ad thus ofte igored for simplicity (though implemeted i the model). With two equally-probable categories (as is most commo i category learig paradigms) ) is also costat (.5), ad thus the mai determiat of classificatio is F i ). This probability is calculated i the followig maer: i (2)

2 where F i F i + α Fi Fi ) = (3) + α is the umber of items with feature F i i category, c is the total umber of items i category, ad α F i ad α are uiform priors. Equatio 3 ca be iterpreted as updatig a uiform prior with ew iformatio, with the prior evetually overwhelmed as more features are observed. The classifier ca be exteded to reflect uderlyig depedecies betwee features that are ot idepedetly geerated. This refiemet ca ofte be useful i categorizatio studies whe oe feature costrais the values of other features. Whe these depedecies are kow, they ca be icorporated ito the model by retaiig the relevat coditioal probabilities. For example, Equatio 4 is a toy model with two features i which feature 2 is depedet o feature (the ormalizatio costat is left out for simplicity): p F, F ) = ) F ) F, ) (4) ( 2 2 F Equatio 3 ca also be exteded for features that are depedet o other features, becomig: F i, F + α j Fi Fi, F j ) = (5) + α where F i is depedet o feature F j. The algorithm for category learig operates as follows. First, a ew example is preseted to the model without category label iformatio. The probability of its beig i each category is calculated based o previously-observed labeled examples, ad the resultig probabilities are used to assig a predicted category to the example 2. The typical way to classify a ew example is to choose the category with the maximum probability of geeratig the example (Geisler, 2003). Oce category assigmet is complete the example is placed ito the observed set alog with its category label. This step simulates the effect of feedback, with the ew example ow affectig future classificatio judgmets. Order of presetatio is importat: like the participats, the model s predictios ca oly be based o previously see exemplars. There are may ways to compare the model s performace with humas. Oe possibility is to use a metric based o the umber of correct ad icorrect trials, which is perhaps the closest aalog to how ideal observers have bee used i recet studies (Geisler, 2003). However, may More specifically, the α Fi s describe the parameters of a Dirichlet prior i which all values are set to, with their sum beig α c. 2 Oe issue with the algorithm is how to get it started. Although there are a umber of justifiable methods, here we start the model with the smallest umber of examples for which there is oe example for each category. category learig paradigms focus o differeces i learig rates, with a commo metric beig the umber of trials eeded to reach a certai performace criterio. Viewed i terms of statistical samplig (e.g., the umber of samples eeded to lear a certai distributio), this metric provides a atural compariso of huma ad model learig. Specifically, we ca defie samplig efficiecy as the ratio of the umber of trials the model eeded to lear to criterio to the umber of trials a huma eeded (see Scholkopf & Smola, 2002; Stakiewitz, 2003): ttc mod (6) ttc par where ttc mod is the trials to criterio eeded for the model ad ttc par the trials eeded for the participat. The closer huma performace comes to the ideal, the higher the efficiecy. I may ways the ideal observer described here is similar to Aderso s (99) ratioal theory of categorizatio. However, Aderso focuses o determiig the optimal categorizatio give a geeral eviromet i pursuit of a descriptive theory of huma categorizatio. I cotrast, our ideal observer simply aims to be ormative i a specific eviromet for which the structure ad geeratig model is kow, ad to provide a bechmark or upper boud o huma performace. Thus may of the goals ad assumptios of Aderso s model are very differet from the ideal observer described here. For example, sice we kow ad capture the depedecies betwee features, we do ot make the simplifyig assumptio that all features are coditioally idepedet. Droppig this assumptio is ecessary i order to maitai optimality for the types of geeratig models commoly used i higher-order category learig, where features are ofte costraied by the values of other relatios or features. Also, istead of predictig a usee feature (such as the category label) through chaied iferece, we focus o the simpler task of predictig a category class give a set of features. This simpler goal allows us to avoid usig weighted category averages ad oly requires computatio of the maximum likelihood category. Fially, we avoid the eed for a empirically-determied variable goverig the probability of creatig a ew category (Aderso s couplig probability ). We will ow apply this ideal observer i order to model learig i a categorizatio experimet i which the differet coditios may have differet types ad amouts of iformatio associated with them. The Experimet A fudametal shift i the uderstadig of categorizatio resulted from the family resemblace view of categories, which argued that may categories have a graded structure based o shared features (Medi & Schaffer, 978; Rosch, 976; Rosch & Mervis, 975; Wittgestei, 953). The family resemblace view has had great success accoutig for peoples learig ad geeralizatio of categories that ca be represeted as simple lists of features. Such

3 categories ca be leared implicitly ad automatically, with feature-category associatios ot ecessarily available to coscious verbalizatio (Ashby, Maddox, & Bohil, 2002; Ashby & Waldro, 999). However, much of huma coceptual kowledge is composed of categories that caot be represeted as simple features (Barsalou, 983; Keil, 989; Murphy & Medi, 985; Rips, 989; Ross & Spaldig, 994). Rather, may cocepts are based o the relatioships betwee thigs rather tha the literal features of the thigs themselves. For example, a barrier is a relatioal cocept that ca be as cocrete as a wall or moat or as abstract as a lack of moey or the color of oe s ski. Relatioal cocepts aboud i everyday life, with examples icludig social uderstadig (a love triagle), law (breach of cotract), religio (atoemet for sis), sciece (coservatio of eergy), as well as basic perceptio (recogizig arragemets of objects as scees) (e.g., Geter & Kurtz, 2005, Holyoak & Thagard, 995). Although relatioal cocepts are fudametal to huma itelligece, our uderstadig of how we lear them is poor compared to our uderstadig of feature-based categories. A reasoable ad parsimoious hypothesis is that relatioal categories act just like feature-based categories with the features replaced by relatios that is, cocept learig may be a sigle uified process that ca take either features or relatios as iput. This view predicts that relatioal categories should show the same kid of family resemblace structure evideced by feature-based categories, thus geeralizig what we have leared about category learig from feature-based to relatioal categories. However, there is evidece that relatios ad features may be psychologically distict. For example, Medi, Goldstoe, ad Geter (990, 993) demostrated strog empirical differeces betwee relatioal ad feature-based similarity, suggestig that relatios ad features may rely o separate, competig processes for assessig similarity. osistet with these fidigs, some researchers have argued that feature lists are fudametally iadequate to represet relatioal cocepts, ad that such cocepts must istead be metally represeted as relatioal structures such as schemas or theories (Geter, 983; Hollad, Holyoak, Nibett, & Thagard, 986; Hummel & Holyoak, 2003; Keil, 989; Murphy & Medi, 985). I such accouts, learig a relatioal category is more aki to iducig a schema tha to learig a list of diagostic features. Most accouts of schema iductio assume that a shared, determiistic cohesive elemet is ecessary to create the schema i the first place (Hummel & Holyoak, 2003; Kuehe et al., 2000). We coducted a experimet to test whether relatioal ad feature-based categories were leared i similar ways 3. Specifically, we hypothesized that relatioal categories i which o sigle defiig elemet existed as is the case i family resemblace categories would prove drastically more difficult to lear tha feature-based categories with a 3 The experimet described here is based o pilot data reported i Kittur, Hummel, ad Holyoak (2004), which describes i more detail the methods used ad additioal measures collected. idetical family resemblace structure. Whereas learig family resemblace categories based o simple features may be doe implicitly through trackig ad averagig the features of the exemplars of each category, learig relatioal categories will be much more difficult because the same feature(s) may be associated with multiple categories, depedig o the relatios ivolved. To test this hypothesis we used a 2x2 betwee-subjects desig, i which categories either had a sigle dimesio perfectly predictive of category membership (determiistic) or had a family resemblace structure i which three out of four dimesios were characteristic of the category but o sigle dimesio was perfectly predictive. Dimesios were defied either by idividual feature values or by the relatios betwee features. We predicted a iteractio betwee category structure ad type, such that the relatioal family resemblace coditio would be much more difficult to lear tha ay of the other three coditios. Method Subjects. 96 Uiversity of aliforia, Los Ageles udergraduates participated for partial fulfillmet of course requiremets. Figure. Examples of family resemblace categories. Determiistic relatioal categories were formed by removig oe exemplar from each category. The table depicts the dimesios categories were defied o, as well as the value of each exemplar o the dimesios (filled = value, empty = value 2). For example, i relatioal category exemplar i, the octago (O) is bigger, darker, above, ad behid the square (S). Show are oly a small subset of all istatiatios of the four exemplar types for a category.

4 Stimuli ad Procedure. All stimuli were composed of a octago ad a square set i a fixed backgroud resemblig a computer chip. Either the relatios betwee the two shapes (relatioal coditio) or the idividual features of each shape (feature-based coditio) determied category membership of each exemplar. Relatioal categories were defied by whether the octago was ) larger, 2) darker, 3) vertically above, ad 4) i frot of the square (see Figure ). Feature-based categories were defied by idividual absolute feature values: ) size of the octago, 2) color of the octago, 3) size of the square, ad 4) color of the square. rossed with the feature-based ad relatioal coditios was the structure of each category. I the family resemblace coditio, each category member had three out of four dimesios cosistet with its category ad oe icosistet dimesio. I the determiistic coditio oe dimesio was perfectly diagostic across all exemplars. This desig yielded four coditios to which participats were radomly assiged: relatioal family resemblace or determiistic (R-FR or R-D) ad feature-based family resemblace or determiistic (F-FR or F-D). O each trial of the acquisitio phase, a participat viewed oe exemplar, categorized it as a math or graphics chip, ad received accuracy feedback. Acquisitio cotiued util the participat reached criterio (>88% correct for two cosecutive blocks 4 ). Behavioral Results The relatioal family resemblace coditio proved much more difficult to lear tha the other three coditios: 22% of participats i the relatioal family resemblace coditio did ot lear to criterio withi 600 trials (o participats i ay other coditios failed to lear). All results make the extremely coservative assumptio that participats who failed to lear would have succeeded o trial 60. The mea umber of trials to criterio for each coditio is show i Figure 2a. There were mai effects of both category type (relatios vs. features, F(, 95) = 4.7, p =.032, ad category structure (family resemblace vs. determiistic, F(, 95) = 9.83, p =.002; importatly, there was also a sigificat iteractio of category type ad structure, F(,95) = 6.4, p =.05, due to extremely impaired acquisitio whe the category was defied by relatios ad had a family resemblace structure. Ideal Observer Aalysis Oe explaatio of these results is that relatios ad features are represeted ad processed differetly i the brai, ad that relatioal categories may ot have access to the machiery that is used to lear feature-based family resemblace categories. However, aother explaatio could be that the selective impairmet of the relatioal family resem- 4 This criterio was chose so that simple feature-trackig strategies (e.g., memorizig the associatios of sigle features with categories) would lead to sub-criterio performace. Trials to riterio (Huma) Trials to riterio (Model) Efficiecy (Model/Huma) Featural Featural Featural Det FR (a) Relatioal Relatioal Relatioal (b) Figure 2. (a) Mea trials to criterio take by participats to lear the categories. Det=Determiistic, FR=Family Resemblace. (b) Mea trials to criterio for the model to lear the categories give the same stimuli i the same order as each idividual participat. (c) Efficiecy of huma performace compared to model performace measured by the ratio of the umber of trials eeded by the model to lear to criterio to the umber of trials eeded by huma learers. (Note that efficiecy is calculated o a per-subject basis, ad so caot be determied from paels (a) ad (b) aloe.) blace coditio is istead due to a differece i the amout of available diagostic iformatio. I other words, are people worse oly because some coditios are iheretly more difficult to lear due to lack of iformatio? To aswer this questio we adapted the ideal observer model described earlier to the curret experimetal task. The features ad relatios available to participats were coded as discrete values o separate dimesios ad used as iputs to the model. For example, the model received as separate iputs the size of the square, the size of the octago, ad the relatio of which was bigger. The same iformatio was available to participats, who could use iformatio about either the features or the relatios o each trial. However, the relatioal ad featural iformatio o a dimesio were ot idepedet: i the example above, if the relatio was octago bigger tha square, the kowig the size of the octago provides iformatio about the size of the square (which must be smaller; see Figure 3). To accout for this depedecy, relatios were modeled as idepedet (c)

5 Figure 3. Relatios ad features ivolved i category geeratio. Arrows depict depedecies (i.e., costraits) betwee relatios ad features. iputs, whereas feature iputs were coditioal o their respective relatio. 5 For the ideal observer described here to be truly optimal, the geeratig model must meet certai assumptios. First, the distributio of category members must be sampled from idepedet multiomial distributios with Dirichletdistributed parameters. This assumptio holds true: category members were geerated by samplig from idepedet multiomial distributios with a equal likelihood of each member appearig. Secod, all depedecies that arise i geeratig feature values for each category member must be captured i the coditioal probabilities (i.e., relatios costraiig features). This assumptio is also valid: the depedecies show i Figure 3 reflect how the exemplars were geerated. The ideal observer model was ru o each participat s data, ad the umber of trials ecessary to lear to criterio was measured 6 (see Figure 2b). We the computed each participat s efficiecy accordig to Equatio 6. These efficiecies are depicted i Figure 2c. Huma performace as measured by statistical efficiecy was much worse i the relatioal family resemblace coditio tha i ay other coditio. A ANOVA was performed o the efficiecy measure followig a log trasformatio to ormalize the variaces. The results demostrated that the critical iteractio was sigificat, F(, 95) = 3.93, p <.05. Sice efficiecy takes ito accout differeces i model as well as huma performace, fidig a iteractio o this measure idicates that the huma learig rates for these coditios were more differet tha would expected give the iheret difficulty of the coditios. That is, iheret iformatioal differeces betwee the coditios were isufficiet to accout for the disparities i huma performace. Discussio The behavioral results revealed a clear impairmet i acquisitio for relatioal categories defied by a family resemblace structure, as compared to categories based o features, which are leared quickly whether they had family resemblace or determiistic structure. Relatioal catego- 5 A atural questio is: should the features be defiig i the featural coditios, rather tha the relatios? No chage is eeded i the model because i the featural coditios the dimesios o which the features were cosidered depedet (relative size ad shade) had the same relatioal value for both categories. Thus the features become effectively idepedet. 6 A more statistically accurate phrasig of this would read: the umber of samples eeded to lear the distributio to a certai degree of accuracy. ries with determiistic structure were leared as quickly as determiistic feature-based categories, suggestig that the effect is ot merely due to the relatioal ature of the task. This iteractio is icosistet with the hypothesis that relatioal categories are leared i the same way as featurebased categories. A ideal observer aalysis was used to determie whether this impaired learig might be due to iheret iformatioal differeces betwee coditios. By comparig the efficiecy of huma performace to that of the ideal model, we were able to show that objective differeces i difficulty betwee coditios did ot accout for the experimetal data. Rather, it appears that relatios ad features are represeted or processed differetly i huma category learig. Idetifyig exactly how relatios ad features differ is a importat subject for future research. Oe potetially useful approach is to determie what chages to the ideal observer could make it more closely match huma data. For example, what happes whe the model does ot have perfect memory, or caot perfectly update its prior? Or whe its workig memory is impaired so it caot atted to all relatios ad features at oce? Observig how the model degrades as additioal costraits are added could provide valuable isights ito huma iformatio processig. Alteratively, it is possible that o processig-related chages i the ideal observer will capture the dissociatio i the huma data. Istead, it may be ecessary to take ito accout the represetatioal differece betwee simple features ad relatioal predicates. It remais a ope questio how to icorporate structured predicates ito a Bayesia framework; extat aalyses of categorizatio usig Bayesia iferece treat relatios as correlatios or ustructured features rather tha as structured predicates (e.g., Kemp et al., 2004). Ideed, oe possible explaatio of our results may be that the likelihood updatig mechaism at work i featural categorizatio may ot be used for relatioal categorizatio, resultig i impairmet of relatioal family resemblace learig. At first glace the preset results appear couterituitive: relatioal category learig is severely impaired if o elemets are costat across all exemplars, yet people seem able to coceptualize family resemblace relatioal categories, such as Wittgestei s (953) classic example of game. This paradox highlights the eed for additioal empirical studies. Oe approach to explorig this seemig icosistecy may be to examie prior kowledge ad experiece. While a sigle costat elemet may be ecessary to lear ovel relatioal categories, whe prior kowledge ad experiece are brought ito play this critical eed may be reduced. It is possible that a coheret theory that explais a relatioal family resemblace structure might make learig easier (Rehder & Hastie, 2004; Rehder & Ross, 200). I additio, repeated experiece with the relevat relatios may lead to low-level chukig of a stimulus, as i chess experts memory for board positios. Thus both higher-order causal explaatios ad lower-order experiece may facilitate relatioal learig.

6 I summary, the dissociatio betwee feature-based category learig, which is robust to family-resemblace structure, ad relatio-based category learig, which is ot, suggests that curret feature-based models of category learig may have limited applicability to relatioal categories. The difficulty for such models is ot oly that feature lists are iadequate to represet relatios, but that the two kids of categories are processed differetly as well. The preset study also demostrates how ideal observer methods ca be applied i higher-order category learig. Here we used a ideal observer to provide a objective measure of the ease of learig i each coditio. The model is easy to implemet i category learig studies with discrete stimuli for which the geeratig model is kow. We believe that the ideal-observer approach ca have geeral applicability for studies of category learig i which differet learig coditios may have differet iformatioal cotet. Ackowledgmets Preparatio of this paper was supported by NSF grat SES to KH. Special thaks to Hogjig Lu, Zili Liu, ad Barbara Spellma for helpful commets ad discussio. Refereces Aderso, J. R. (99). The adaptive ature of huma categorizatio. Psychological Review, 98, Ashby, F. G., Maddox, W. T., & Bohil,. J. (2002). Observatioal versus feedback traiig i rule-based ad iformatio-itegratio category learig. Memory & ogitio, 30, Ashby, F. G., & Waldro, E. M. (999). O the ature of implicit categorizatio. Psychoomic Bulleti & Review, 6, Barsalou, L. W. (983). Ad hoc categories. Memory & ogitio,, Domigos, P., & Pazzai, M. J. (997). O the optimality of the simple Bayesia classifier uder zero-oe loss. Machie Learig, 29, Geisler, W. S. (2003). Ideal Observer aalysis. I L. halupa, ad J. Werer (Eds.), The visual euroscieces (pp ). Bosto: MIT Press. Geter, D., & Kurtz, K. J. (2005). Learig ad usig relatioal categories. I W. K. Ah, Goldstoe, R.L., Love, B.., Markma, A.B., & Wolff, P.W. (Ed.), ategorizatio iside ad outside the lab. Washigto, D..: APA. Geter, D. (983). Structure-mappig: A theoretical framework for aalogy. ogitive Sciece, 7, Hollad, J. H., Holyoak, K. J., Nisbett, R. E., & Thagard, P. R. (986). Iductio: Processes of Iferece, Learig, ad Discovery. ambridge, MA, US: The MIT Press. Holyoak, K. J., & Thagard, P. (995). Metal leaps: Aalogy i creative thought. ambridge, MA: The MIT Press. Hummel, J. E., & Holyoak, K. J. (2003). A symboliccoectioist theory of relatioal iferece ad geeralizatio. Psychological Review, 0, Keil, F.. (989). ocepts, kids, ad cogitive developmet. ambridge: The MIT Press. Kemp,., Griffiths, T. L., & Teebaum, J. B. (2004). Discoverig latet classes i relatioal data. MIT AI Memo Kuehe, S., Forbus, K. D., Geter, D., & Qui, B. (2000). SEQL: ategory learig as progressive abstractio usig structure mappig. Proceedigs of the Twety-secod Aual oferece of the ogitive Sciece Society. Legge, G. E., Klitz, T. S., & Tja, B. S. (997). Mr. hips: A ideal-observer model of readig. Psychological Review, 04, Liu, Z., Kill, D.., & Kerste, D. (995). Object classificatio for huma ad ideal observers. Visio Research, 35, Medi, D. L., Goldstoe, R. L., & Geter, D. (990). Similarity ivolvig attributes ad relatios: Judgmets of similarity ad differece are ot iverses. Psychological Sciece,, Medi, D. L., Goldstoe, R. L., & Geter, D. (993). Respects for similarity. Psychological Review, 00, Medi, D. L., & Schaffer, M. M. (978). otext theory of classificatio learig. Psychological Review, 85, Murphy, G. L. (2002). The big book of cocepts. ambridge: The MIT Press Murphy, G. L., & Medi, D. L. (985). The role of theories i coceptual coherece. Psychological Review, 92, Rehder, B., & Hastie, R. (2004). ategory coherece ad category-based property iductio. ogitio, 9, Rehder, B., & Ross, B. H. (200). Abstract coheret categories. Joural of Experimetal Psychology: Learig, Memory, & ogitio, 27, Rips, L. J. (989). Similarity, typicality, ad categorizatio. I S. O. Vosiadou, Adrew (Ed.), Similarity ad aalogical reasoig. New York: ambridge Uiversity Press. Rosch, E., & Mervis,. B. (975). Family resemblaces: Studies i the iteral structure of categories. ogitive Psychology, 7, Rosch, E., Simpso,., & Miller, R. S. (976). Structural bases of typicality effects. Joural of Experimetal Psychology: Huma Perceptio & Performace, 2, Ross, B. H., & Spaldig, T. L. (994). ocepts ad categories. I R. J. Sterberg (Ed.), Thikig ad problem solvig. Hadbook of perceptio ad cogitio (2d ed.) (pp. 9-48). Sa Diego, A: Academic Press, Ic. Scholkopf, B., & Smola, A. J. (2002). Learig with Kerels: MIT Press. Stakiewicz, B.J., Legge, G.E., Masfield, J.S., & Schlicht, E.J. (i press). Lost i virtual space: Studies i huma ad ideal spatial avigatio. Joural of Experimetal Psychology: Huma Perceptio ad Performace. Wittgestei, L. (953). Philosophical ivestigatios. New York: Macmilla.