Categories and Concepts - Spring 2019 Computational models ... · Dalmatians; given three different dogs (or three different animals), adults generalized to all and only the other

Categories and Concepts - Spring 2019Computational models (part 2)

PSYCH-GA 2207

Brenden Lake

where are the others?

Concept learning from just a few positive examples

where are the others?


Concept learning from just a few positive examples“tufa”

(Example from Josh Tenenbaum)

“tufa”

“tufa”

“tufa”


�tufa� �tufa�

�tufa�

Modeling word learning (Tenenbaum & Xu) Concept learning from just a few positive examples

Word Learning as Bayesian Inference

Fei XuUniversity of British Columbia

Joshua B. TenenbaumMassachusetts Institute of Technology

The authors present a Bayesian framework for understanding how adults and children learn the meaningsof words. The theory explains how learners can generalize meaningfully from just one or a few positiveexamples of a novel word’s referents, by making rational inductive inferences that integrate priorknowledge about plausible word meanings with the statistical structure of the observed examples. Thetheory addresses shortcomings of the two best known approaches to modeling word learning, based ondeductive hypothesis elimination and associative learning. Three experiments with adults and childrentest the Bayesian account’s predictions in the context of learning words for object categories at multiplelevels of a taxonomic hierarchy. Results provide strong support for the Bayesian account over competingaccounts, in terms of both quantitative model fits and the ability to explain important qualitativephenomena. Several extensions of the basic theory are discussed, illustrating the broader potential forBayesian models of word learning.

Keywords: word learning, Bayesian inference, concepts, computational modeling

Learning even the simplest names for object categories presentsa difficult induction problem (Quine, 1960). Consider a typicaldilemma faced by a child learning English. Upon observing acompetent adult speaker use the word dog in reference to Max, aparticular Dalmatian running by, what can the child infer about themeaning of the word dog ? The potential hypotheses appear end-less. The word could refer to all (and only) dogs, all mammals, allanimals, all Dalmatians, this individual Max, all dogs plus theLone Ranger’s horse, all dogs except Labradors, all spotted things,all running things, the front half of a dog, undetached dog parts,things that are dogs if first observed before next Monday but catsif first observed thereafter, and on and on. Yet despite this severe

underdetermination, even 2- or 3-year-olds seem to be remarkablysuccessful at learning the meanings of words from examples. Inparticular, children or adults can often infer the approximate ex-tensions of words such as dog given only a few relevant examplesof how the word can be used and no systematic evidence of howwords are not to be used (Bloom, 2000; Carey, 1978; Markman,1989; Regier, 1996). How do they do it?

Two broad classes of proposals for how word learning workshave been dominant in the literature: hypothesis elimination andassociative learning . Under the hypothesis elimination approach,the learner effectively considers a hypothesis space of possibleconcepts onto which words will map and (leaving aside for nowthe problem of homonyms and polysemy) assumes that each wordmaps onto exactly one of these concepts. The act of learningconsists of eliminating incorrect hypotheses about word meaning,on the basis of a combination of a priori knowledge and observa-tions of how words are used to refer to aspects of experience, untilthe learner converges on a single consistent hypothesis. Somelogically possible hypotheses may be ruled out a priori becausethey do not correspond to any natural concepts that the learnerpossesses—for example, the hypothesis that dog refers to thingsthat are dogs if first observed before next Monday but cats if firstobserved thereafter. Other hypotheses may be ruled out becausethey are inconsistent with examples of how the word is used—forexample, one can rule out the hypothesis that dog refers to all andonly cats, or all and only terriers, upon seeing the example of Maxthe Dalmatian.

Settling on one hypothesis by eliminating all others as incorrectamounts to taking a deductive approach to the logical problem ofword learning, and we sometimes refer to these approaches asdeductive approaches. Hypothesis elimination has its roots in earlyaccounts of human and machine concept learning (Bruner, Good-now, & Austin, 1956; Mitchell, 1982), and it corresponds to one ofthe standard paradigms considered in formal analyses of naturallanguage syntax acquisition (Gold, 1967; Pinker, 1979). It is alsorelated to classic inferential frameworks that have been considered

The authors contributed equally to this work and are listed in order ofbirth. This research was made possible by a grant from Mitsubishi ElectricResearch Labs and the Paul E. Newton Chair to Joshua B. Tenenbaum andby a Canada Research Chair and grants from the National Science Foun-dation and the Natural Science and Engineering Research Council ofCanada to Fei Xu.

We thank Eliza Calhoun and Sydney Goddard for assistance withbehavioral experiments. We thank Dare Baldwin, Paul Bloom, Jeff Elman,Jesse Snedeker, and Elizabeth Spelke for helpful discussions about thiswork and Nick Chater, Susan Carey, and Terry Regier for valuable com-ments on the manuscript. We also thank the parents and children for theirparticipation. Portions of this work were presented at the Annual Confer-ence of the Cognitive Science Society (August 2000, Philadelphia, Penn-sylvania; July 2005, Stresa, Italy), the Boston University Conference onLanguage Development (November 2000, Boston, Massachusetts; Novem-ber 2002, Boston, Massachusetts), and the Biennial Conference of theSociety for Research in Child Development (April 2001, Minneapolis,Minnesota).

Correspondence concerning this article should be addressed to Fei Xu,Department of Psychology, University of British Columbia, 2136 WestMall, Vancouver, British Columbia V6T 1Z4, Canada, or to Joshua B.Tenenbaum, Department of Brain and Cognitive Sciences, MassachusettsInstitute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139. E-mail: [email protected] or [email protected]

Psychological Review Copyright 2007 by the American Psychological Association2007, Vol. 114, No. 2, 245–272 0033-295X/07/$12.00 DOI: 10.1037/0033-295X.114.2.245

245

Review: Bayes’ rule for updating beliefs in light of data

Data (D): John is coughing

h1 = John has a coldh2 = John has emphysemah3 = John has a stomach flu

Hypotheses:priorlikelihoodposterior

P (hi|D) =P (D|hi)P (hi)Pj P (D|hj)P (hj)

“Bayes’ rule”

Prior favors h1 and h3, over h2

Likelihood favors h1 and h2, over h3

Posterior favors h1, over h2 and h3

Bayesian concept learning

p(h |X) = P(X | h )P(h )P(X)

h ∈ H : hypothesis about meaning of word (e.g., node in tree structure)X : data (often just labels of positive examples)

�tufa� �tufa�

�tufa�

Modeling word learning (Tenenbaum & Xu)

h 1

h 2 h 3 …

X

Posterior over word meanings

P(X | h ) = [ 1size(h ) ]n

Likelihood

(“the size principle”, uniform random sample from appropriate set)

Prior

P(h )could be uniform over nodes tree, or favor more distinctive nodes,

or favor nodes at a certain level….

n : number of examples

Xu and Tenenbaum Exp 1: Adults

feps from the test set of 24 objects, by clicking on-screen with thecomputer mouse. The test items were laid out in a 4 ! 6 array,with the order randomly permuted from trial to trial.

The experiment began with participants being shown all 24 testobjects, one at a time for several seconds each, to familiarize themwith the stimuli. This familiarization was followed by the instruc-tions and 12 experimental trials. (Some participants were thengiven an additional set of trials, which are not reported here.) Onthe first three trials, participants saw only one example of each newword (e.g., “Here is a fep”). On the next nine trials, they saw threeexamples of each new word (e.g., “Here are three feps”). Withineach set of trials, the example sets appeared in a pseudorandomorder, with content domain (animal, vegetable, and vehicle) andspecificity (subordinate, basic, and superordinate) counterbalancedacross participants. On each trial, the participants were asked tochoose the other objects that the word applied to (e.g., the otherfeps), and their responses were recorded. This phase last approx-imately 15 min in total.

The second phase of the experiment was a similarity judgmenttask. Participants were shown pictures of pairs of objects from theword-learning task and were asked to rate the similarity of the twoobjects on a scale of 1 (not similar at all) to 9 (extremely similar).They were instructed to base their ratings on the same aspects ofthe objects that were important to them in making their choicesduring the word-learning phase. This instruction, along with theplacement of the similarity judgment task after the word-learningtask, was adopted in the hope of maximizing the information thatsimilarity judgments would provide about the hypothesis spacethat participants used in word learning. Similarity judgments took

approximately 45 min to collect. Judgments were collected for allpairs of 39 out of 45 objects—13 from each domain of animals,vegetables, and vehicles—including all test objects and all but 6 ofthe training objects (which were omitted to save time). The 6omitted objects (2 green peppers, 2 yellow trucks, and 2 Dalma-tians) were each practically identical to 3 of the 39 includedobjects, and each was treated as identical to one of those 39 inconstructing the model of learning reported below. Each partici-pant rated the similarity of all pairs of animals, vegetables, andvehicles (78 ! 3 judgments), along with one third of all possiblecross-superordinate pairs (animal–vegetable, vegetable–vehicle,etc.) chosen pseudorandomly (169 judgments), for a total of 403judgments per participant. The order of trials and the order ofstimuli were randomized across participants. These trials werepreceded by 30 practice trials (chosen randomly from the samestimuli), during which participants were familiarized with therange of similarities they would encounter and were encouraged todevelop a consistent way of using the 1–9 rating scale. They werealso encouraged to use the entire 1–9 scale and to spread theirjudgments out evenly across the scale. The ratings were recorded,and the average rating for each pair of objects was computed.

Results

The main results of Experiment 1 are shown in Figure 5. Adultsclearly differentiated the one-example and the three-example trials,and they were sensitive to the span of the three examples. With oneexample, adults showed graded generalization from subordinate tobasic-level to superordinate matches. These generalization gradi-

Figure 3. Twelve training sets of labeled objects used in Experiment 1, drawn from all three domains (animals,vegetables, and vehicles) and all four test conditions (one example, three subordinate examples, three basic-levelexamples, and three superordinate examples). The circled number underneath each object is used to index thatobject’s location in the hierarchical clustering shown in Figure 7.

254 XU AND TENENBAUM

Helping Mr. Frog who speaks a different language pick out the objects he wants.

“Here is a fep”

ents dropped off more steeply at the basic level, with a softthreshold: Most test items from the same basic-level category werechosen, but relatively few superordinate matches were chosen.With three examples, adults’ generalizations sharpened into amuch more all-or-none pattern. Generalizations from three exam-ples were almost always restricted to the most specific level thatwas consistent with the examples: For instance, given three Dal-matians as examples of feps, adults generalized only to otherDalmatians; given three different dogs (or three different animals),adults generalized to all and only the other dogs (or other animals).

With the above overview in mind, we turn to statistical analysesthat quantify these effects. Later we present a formal computa-tional model of this word-learning task and compare it with the

data from this experiment in more quantitative detail. All analysesin this section were based on one-tailed t tests with plannedcomparisons based on the model’s predictions. Data were col-lapsed over the three different superordinate categories and overthe different test items within a given level of generalization(subordinate, basic, and superordinate). For each of the four kindsof example sets (one, three subordinate, three basic-level, threesuperordinate) and each of the three levels of generalization, eachparticipant received a set of percentage scores measuring howoften he or she had chosen test items at that level of generalizationgiven that kind of example set. The means of these scores acrossparticipants are shown in Figure 5. Because participants almostnever (less than 0.1% of the time) chose any distractors (test itemsoutside of an example’s superordinate category), subsequent anal-yses did not include these scores.

Two questions were addressed with planned t tests. First, didparticipants generalize further in the one-example trials comparedwith the three-example subordinate trials when they were givenone versus three virtually identical exemplars? More specifically,did adults show a significant threshold in generalization at thebasic level in the one-example trials, and did they restrict theirgeneralization to the subordinate level in the three-example trials?Second, did the three-example trials differ from each other de-pending on the range spanned by the examples? More specifically,did participants restrict their generalization to the most specificlevel that was consistent with the set of exemplars?

To investigate the first question, we compared the percentagesof responses that matched the example(s) at the subordinate, basic,and superordinate levels. On the one-example trials, participants

Figure 4. The test set of 24 objects used to probe generalization of word meanings in Experiment 1. For eachtraining set in Figure 3, this test set contains two subordinate matches, two basic-level matches, and foursuperordinate matches. The circled number underneath each object is used to index that object’s location in thehierarchical clustering shown in Figure 7.

Figure 5. Adults’ generalization of word meanings in Experiment 1,averaged over domain. Results are shown for each of four types of exampleset (one example, three subordinate [sub.] examples, three basic-levelexamples, and three superordinate [super.] examples). Bar height indicatesthe frequency with which participants generalized to new objects at variouslevels. Error bars indicate standard errors.

255WORD LEARNING AS BAYESIAN INFERENCE

Which others are feps?

Xu and Tenenbaum Exp 1


“Here are three feps”









Which others are feps?





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Xu and Tenenbaum Exp 1: Training set





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Xu and Tenenbaum Exp 1: Test set









Adult results (Exp 1)













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• Graded generalization with one subordinate example, and rule-like generalization with 3 examples

• Rapid learning from only sparse, positive examples

Xu and Tenenbaum Exp 3: Children





Results





“Here is a fep”









Which others are feps?(YES/NO for each)

• Can 3-4 year olds learn from sparse, positive examples?

• If only one example, it is labeled 3 times to control number of labeling events













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255WORD LEARNING AS BAYESIAN INFERENCEXu and Tenenbaum Exp 3

pick out the others like the ones he has picked out, okay? [Child says“Okay.”]

Three novel words were used: blick, fep, and dax.One-example condition. On each trial, the experimenter

picked out an object from the array (e.g., a green pepper) andlabeled it: “See? A blick.” Then the child was told that Mr. Frogis very picky. The experimenter said to the child, “Now, Mr. Frogwants you to pick out all the blicks from his toys, but he doesn’twant anything that is not a blick. Remember that Mr. Frog wantsall the blicks and nothing else. Can you pick out the other blicksfrom his toys?” The child was then allowed to choose among the24 test objects to find the blicks and put them in front of Mr. Frog.If a child picked out only one object, the experimenter remindedhim or her, “Remember Mr. Frog wants all the blicks. Are theremore blicks?” If a child picked out more than one object, nothingmore was said to encourage him or her to pick out more toys. Atthe end of each trial, the experimenter said to the child, “Now, let’sput all the blicks back and play the game again. Mr. Frog is goingto pick out some more toys, and he would like you to help him pickout others like the ones he picks, okay?” Then another novel wordwas introduced as before.

Each child participated in three trials, each with an exampledrawn from one of the three superordinate categories: a Dalmatian(animal), a green pepper (vegetable), or a yellow truck (vehicle).The order of the trials and the novel words used (blick, fep, anddax) were counterbalanced across participants.

Three-example condition. On each trial, the procedure was thesame as in the one-example trial with the following importantdifference. The experimenter first picked out one object and la-beled it for the child (e.g., “See? A fep.”). Then she picked out twomore objects, one at a time, and labeled each one for the child (e.g.,“Look, another fep” or “Look, this is a fep”). Three factors—thesuperordinate category (animal, vegetable, and vehicle), the rangespanned by the examples (subordinate, basic, and superordinate),and the novel word used (blick, fep, and dax)—were crossedpseudorandomly and counterbalanced across participants. Eachlevel of each factor appeared equally often in the first, second, andthird trials of the experiment.

Results

The patterns of generalization found were qualitatively similarto those found with adults in Experiment 1, and the quantitative

analyses followed essentially the same logic. Analyses were basedon one-tailed t tests with planned comparisons. We collapsedacross superordinate categories, novel words, and trial orders. Foreach type of example set children were shown, they received a setof percentage scores measuring how often they had chosen testitems at each of three levels of generalization (subordinate, basic,and superordinate). The means of these scores across participantsare shown in Figure 6a. Children in the one-example conditioneach received just a single set of scores, because their three trialsall featured the same kind of example set. Children in the three-example condition each received three sets of scores, one for eachtrial, because each trial featured a different kind of example set(three examples clustering at the subordinate, basic, or superordi-nate level). Because no child chose any distractors, subsequentanalyses did not include these scores.

The same two questions as in Experiment 1 were addressed herewith planned t tests. First, did children generalize differently in theone-example trials compared with the three-example trials in eachcase? Of importance, did they generalize differently given oneversus three virtually identical exemplars? More specifically, didchildren show a significant threshold in generalization at the basiclevel in the one-example trials, and did they restrict their general-ization to the subordinate level in the three-example trials? Second,did the three-example trials differ from each other depending onthe range spanned by the examples? More specifically, did chil-dren restrict their generalization to the most specific level that wasconsistent with the set of exemplars?

To investigate the first question, we compared the percentagesof responses that matched the example(s) at the subordinate, basic,and superordinate levels. On the one-example trials, participantschose more subordinate (85%) and basic-level matches (31%) thansuperordinate matches (3%) ( p ! .0001 for both comparisons). Incontrast, when presented with three very similar exemplars fromthe same subordinate category, participants chose more subordi-nate matches (83%) than either basic-level (13%) or superordinatematches (3%) ( p ! .0001 for both comparisons). Similar compar-isons were made between one example and three basic-level orthree superordinate-level examples. When presented with threeexamples from the same basic-level category, participants did notgeneralize more to the basic level as compared with the one-example trials (31% vs. 47%, ns). When presented with threeexamples from the same superordinate category, participants gen-

Figure 6. Children’s generalization of word meanings in Experiments 2 and 3, averaged over domain. Resultsare shown for each of four types of example set (one example, three subordinate [sub.] examples, threebasic-level examples, and three superordinate [super.] examples). Bar height indicates the frequency with whichparticipants generalized to new objects at various levels. Error bars indicate standard errors.


Adults

Children

Bayesian concept learning

p(h |X) = P(X | h )P(h )P(X)

h ∈ H : hypothesis about the meaning of word (node in tree structure)

X : 1 or 3 positive examples


P(X | h ) = [ 1size(h ) ]n ≈ [ 1

height(h ) + ϵ]n

Likelihood

(height is the average within-node distance between examples)

PriorP(h ) ∝ height(parent[h ])) − height(h )

favors more distinctive nodes,or favor nodes at a certain level….

node of the tree corresponds to a cluster of objects that are onaverage more similar to each other than to other nearby objects.The height of each node represents the average pairwise dis-similarity of the objects in the corresponding cluster. The lengthof the branch above each node measures how much moresimilar on average are that cluster’s members to each other thanto objects in the next nearest cluster—that is, how distinctivethat cluster is.

Each of the main classes underlying the choice of stimulicorresponds to a node in the tree: vegetable (EE), vehicle (HH),animal (JJ), pepper (J), truck (T), dog (R), green pepper (F),yellow truck (G), and Dalmatian (D). Most of these clusters arehighly distinctive (i.e., well separated from other clusters bylong branches), as one would expect for the targets of kindterms.2 Other easily describable nodes include Cluster U, con-taining all and only the construction vehicles (tractor, bull-dozer, and crane), and Cluster II, containing all and only the

mammals. The only clusters that do not appear to correspond toconceivably lexicalizable concepts are two that are defined onlyby subtle perceptual variation below the subordinate level:Cluster A, including two of the three Dalmatians, and Cluster B,including two of the three green peppers. We take each clusterto correspond to one hypothesis in H, with the exception ofthese two clusters below the subordinate level. In so doing, weare assuming that each learner maintains only a single hypoth-esis space and that its structure does not change as new words

2 A notable exception is the cluster corresponding to trucks (T), which isbarely separated from the next highest cluster (V), which contains thetrucks plus a long yellow school bus. Cluster V itself is fairly wellseparated from the next highest cluster, suggesting that the perceptuallybasic category here is not quite trucks but something more like “truck-shaped motor vehicles.”

Figure 7. Hierarchical clustering of similarity judgments yields a taxonomic hypothesis space for Bayesianword learning. Letter codes refer to specific clusters (hypotheses for word meaning): vegetable (EE), vehicle(HH), animal (JJ), pepper (J), truck (T), dog (R), green pepper (F), yellow truck (G), and Dalmatian (D). Theclusters labeled by other letter codes are given in the text as needed. Numbers indicate the objects located at each leafnode of the hierarchy, keyed to the object numbers shown in Figures 3 and 4. The height of a cluster, as given by thevertical axis on the left, represents the average within-cluster dissimiliarity of objects within that cluster.


n : number of examples

animalvehicle vegetable

pepper

Generalizing to a new example y

p(y ∈ C |X) = ∑h ∈H

P(y ∈ C | h )p(h |X)













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255WORD LEARNING AS BAYESIAN INFERENCEModel results

Adults

Bayesian model

Model Results

We first consider the basic Bayesian model using the distinc-tiveness prior, Equation 7. Figure 8a compares p(y ! C|X) com-puted from this model with the generalization judgments of ouradult participants (Figures 5 and 8d), averaged across participants,superordinate classes (animal, vehicle, and vegetable), and testitems within a given level of generalization. On the averaged datashown in Figure 8d, the model achieves a reasonable quantitativefit (r ! .89).4 It also captures the main qualitative features of thedata: graded generalization given one example, and more all-or-none, rulelike generalization at the level of the most specificconsistent natural concept given three examples. However, thereare also several differences between the model’s generalizationsand people’s judgments: The model produces too little generali-zation to basic-level matches given one example or three subordi-nate examples and too much generalization to superordinatematches given three basic-level examples.

Figure 8b shows the fit of the Bayesian model after incorporat-ing a bias in the prior that favors the three basic-level hypotheses.The strength of the basic-level bias is a free parameter, here set to" ! 10. With this one free parameter, the model now provides analmost perfect fit to the average data (r ! .99). All of the mainqualitative trends are captured, including those not accounted forby the Bayesian model without a basic-level bias (in Figure 8a).These results suggest that, at least for adults, hypotheses for wordlearning are biased specifically toward basic-level object catego-ries, over and above a general preference for more distinctivecategories that was captured in the branch length prior (Equation 7and Figure 8a).

A different picture emerges when we compare these two ver-sions of the Bayesian model with preschool-age children’s gener-alizations (Experiment 3; Figures 6b and 8c). In some ways,children’s performance looks more like the Bayesian model’spredictions without the basic-level bias, particularly in the shiftfrom one example to three subordinate examples. Correlationcoefficients for the two models are similar (r ! .91 without the

basic-level bias, r ! .89 with the basic-level bias). Because theadditional parameter " does not contribute significantly to thevariance accounted for and leads to a fit that is qualitatively worsein some ways, these results suggest that child word learners maynot have the strong basic-level bias that adults exhibit. Theirtendency to extend new words to basic-level matches is muchweaker than that of adults and may be explained simply as thecombination of Bayesian hypothesis averaging (Equation 3) with ageneral preference for hypotheses corresponding to distinctivecategories (Equation 7). We return to this issue in the discussionbelow.

Comparison With Other Models

Figure 9 illustrates respectively the complementary roles playedby the size principle (Equations 5 and 6) and hypothesis averaging(Equation 3) in the Bayesian framework. If instead of the sizeprinciple we weight all hypotheses strictly by their prior (includingthe basic-level bias), Bayes reduces to a similarity-like feature-matching computation that is much more suited to the generaliza-tion gradients observed given one example than to the all-or-nonepatterns observed after three examples (Figure 9a). Mathemati-cally, this corresponds to replacing the size-based likelihood inEquations 5 and 6 with a simpler measure of consistency: p(X|h) !1 if the examples X are consistent with the hypothesis h (i.e., xi !h for all i) and p(X|h) ! 0 otherwise. Tenenbaum and Griffiths(2001) called this approach weak Bayes, because it uses only aweak binary measure of consistency in the likelihood rather thanthe strong assumption of randomly sampled examples implicit inusing the size principle. Essentially this algorithm has been pro-

4 All correlation (r) values in this section were computed using onlyjudgments for test items within the same superordinate class as the ob-served examples. Participants almost never chose test items that crossedsuperordinate boundaries, and most models give these test items zero ornear-zero probability of generalization.

Figure 8. Predictions of the Bayesian model, both with and without a basic-level bias, compared with the datafrom adults in Experiment 1 and those from children in Experiment 3. Sub. ! subordinate; super. !superordinate.


Alternative Bayesian models

posed by Mitchell (1997), Haussler, Kearns, and Schapire (1994),and Shepard (1987).

If instead of averaging the predictions of all consistent hypoth-eses we base generalization on just the single most probablehypothesis, Bayes reduces to an all-or-none rulelike computation.Priors (again including the basic-level bias) and likelihoods coop-erate to rank hypotheses, but only the highest ranking hypothesis—rather than a probability distribution over hypotheses—is used ingeneralization. Mathematically, this corresponds to replacing hy-pothesis averaging in Equation 3 with a simpler decision rule:p(y ! C|X) ! 1 if y ! h*, and 0 otherwise, where h* is thehypothesis with maximal posterior probability p(h|X) (in Equation2). This approach is called maximum a posteriori Bayes, or MAPBayes for short. As Figure 9b shows, MAP Bayes captures thequalitative trends in how adults and children generalize frommultiple examples, including the restriction of generalization afterthree subordinate examples have been observed.5 However, it doesnot capture the graded nature of generalization from a singleexample. It also does not capture the increasing confidence inbasic-level generalization that comes from seeing three basic-levelexamples; unlike both adults and children, MAP Bayes makesexactly the same generalizations from three basic-level examplesas it does from just a single example.

Figure 10 shows the predictions of four alternative learningmodels. None of these models have been specifically proposed forword learning, but they are generic approaches from the literatureon computational models of learning and generalization, and theyare representative of previous suggestions for how word learningmight be viewed computationally. None are explicitly Bayesian,but to varying degrees they correspond to the two special cases ofBayesian learning shown above. Figure 10a presents the predic-tions of a simple exemplar-similarity model, in which p(y ! C|X)is computed by averaging the similarity of y to each exemplar in X.(We use the mean similarity judgments of the adult participants inExperiment 1, normalized to a 0–1 scale.) For each set of exam-ples, the generalization function is scaled linearly to have a max-imum at 1.

Figure 10b shows the predictions of an alternative approach toexemplar similarity, inspired by proposals of Goldstone (1994) andOsherson, Smith, Wilkie, Lopez, and Shafir (1990), in whichp(y ! C|X) is computed by taking the maximum similarity of y toall exemplars in X. Like weak Bayes, the pure hypothesis-averaging version of the Bayesian model shown in Figure 9a, bothexemplar-similarity models give a soft gradient of generalization

from one example but fail to sharpen generalization to the appro-priate level given three examples.

More flexible similarity-based models of category learning thatincorporate selective attention to different stimulus attributes (e.g.,Kruschke, 1992) might be better able to accommodate our data,but not without major modification. These models typically rely onerror-driven learning algorithms, which are not designed to learnhow broadly they should generalize from just one or a few positiveexamples without any negative examples, and low-dimensionalspatial representations of stimuli, which are not suited to repre-senting a broad taxonomy of object kinds.

Several authors have suggested that associative or correlationallearning algorithms, perhaps instantiated in neural networks, canexplain how children learn the meanings of words (Colunga &Smith, 2005; Gasser & Smith, 1998; Regier, 1996, 2003). It is notpossible here to evaluate all extant correlational learning algo-rithms, but we do consider the standard approach of Hebbianlearning (Hertz, Krogh, & Palmer, 1991). Figure 10c shows thepredictions of a Hebbian learning network that is matched asclosely as possible in structure to our Bayesian models. TheHebbian model uses input features corresponding to the samehypotheses used in our Bayesian models, but instead of evaluatingand averaging those hypotheses with the machinery of Bayesianinference, it uses the Hebb rule to compute associative weightsbetween each input feature unit and an output unit representing theoccurrence of the novel word to be learned (e.g., fep). This net-work produces generalization patterns very much like those pro-duced by the exemplar-similarity models (Figure 10a, 10b) orweak Bayes (Figure 9a), capturing something of the graded char-acter of one-shot generalization but failing to account for howgeneralization sharpens to the appropriate level after three exam-ples are seen.

The similar predictions of these various models reflect twounderlying computational commonalities. First, learning in theHebbian network is strictly based on the frequency with whichinput features occur in the observed examples: Each exemplarleaves a trace of its feature values in the weights connecting inputfeatures to the output unit, and the final pattern of generalization is

5 Figure 9b shows the median pattern of generalization over the threesuperordinate categories rather than the mean because the MAP generali-zations are always either 0 or 1, and thus the mean is sometimes notrepresentative of the model’s all-or-none predictions.

Figure 9. Predictions of two variants of the Bayesian model. (a) Without the size principle, Bayesiangeneralization behaves like an exemplar-similarity computation. (b) Without hypothesis averaging, Bayesiangeneralization follows an all-or-none, rulelike pattern. MAP Bayes ! maximum a posteriori Bayes approach;sub. ! subordinate; super. ! superordinate.


Model Results










posed by Mitchell (1997), Haussler, Kearns, and Schapire (1994),and Shepard (1987).

If instead of averaging the predictions of all consistent hypoth-eses we base generalization on just the single most probablehypothesis, Bayes reduces to an all-or-none rulelike computation.Priors (again including the basic-level bias) and likelihoods coop-erate to rank hypotheses, but only the highest ranking hypothesis—rather than a probability distribution over hypotheses—is used ingeneralization. Mathematically, this corresponds to replacing hy-pothesis averaging in Equation 3 with a simpler decision rule:p(y ! C|X) ! 1 if y ! h*, and 0 otherwise, where h* is thehypothesis with maximal posterior probability p(h|X) (in Equation2). This approach is called maximum a posteriori Bayes, or MAPBayes for short. As Figure 9b shows, MAP Bayes captures thequalitative trends in how adults and children generalize frommultiple examples, including the restriction of generalization afterthree subordinate examples have been observed.5 However, it doesnot capture the graded nature of generalization from a singleexample. It also does not capture the increasing confidence inbasic-level generalization that comes from seeing three basic-levelexamples; unlike both adults and children, MAP Bayes makesexactly the same generalizations from three basic-level examplesas it does from just a single example.

Figure 10 shows the predictions of four alternative learningmodels. None of these models have been specifically proposed forword learning, but they are generic approaches from the literatureon computational models of learning and generalization, and theyare representative of previous suggestions for how word learningmight be viewed computationally. None are explicitly Bayesian,but to varying degrees they correspond to the two special cases ofBayesian learning shown above. Figure 10a presents the predic-tions of a simple exemplar-similarity model, in which p(y ! C|X)is computed by averaging the similarity of y to each exemplar in X.(We use the mean similarity judgments of the adult participants inExperiment 1, normalized to a 0–1 scale.) For each set of exam-ples, the generalization function is scaled linearly to have a max-imum at 1.

Figure 10b shows the predictions of an alternative approach toexemplar similarity, inspired by proposals of Goldstone (1994) andOsherson, Smith, Wilkie, Lopez, and Shafir (1990), in whichp(y ! C|X) is computed by taking the maximum similarity of y toall exemplars in X. Like weak Bayes, the pure hypothesis-averaging version of the Bayesian model shown in Figure 9a, bothexemplar-similarity models give a soft gradient of generalization

from one example but fail to sharpen generalization to the appro-priate level given three examples.

More flexible similarity-based models of category learning thatincorporate selective attention to different stimulus attributes (e.g.,Kruschke, 1992) might be better able to accommodate our data,but not without major modification. These models typically rely onerror-driven learning algorithms, which are not designed to learnhow broadly they should generalize from just one or a few positiveexamples without any negative examples, and low-dimensionalspatial representations of stimuli, which are not suited to repre-senting a broad taxonomy of object kinds.

Several authors have suggested that associative or correlationallearning algorithms, perhaps instantiated in neural networks, canexplain how children learn the meanings of words (Colunga &Smith, 2005; Gasser & Smith, 1998; Regier, 1996, 2003). It is notpossible here to evaluate all extant correlational learning algo-rithms, but we do consider the standard approach of Hebbianlearning (Hertz, Krogh, & Palmer, 1991). Figure 10c shows thepredictions of a Hebbian learning network that is matched asclosely as possible in structure to our Bayesian models. TheHebbian model uses input features corresponding to the samehypotheses used in our Bayesian models, but instead of evaluatingand averaging those hypotheses with the machinery of Bayesianinference, it uses the Hebb rule to compute associative weightsbetween each input feature unit and an output unit representing theoccurrence of the novel word to be learned (e.g., fep). This net-work produces generalization patterns very much like those pro-duced by the exemplar-similarity models (Figure 10a, 10b) orweak Bayes (Figure 9a), capturing something of the graded char-acter of one-shot generalization but failing to account for howgeneralization sharpens to the appropriate level after three exam-ples are seen.

The similar predictions of these various models reflect twounderlying computational commonalities. First, learning in theHebbian network is strictly based on the frequency with whichinput features occur in the observed examples: Each exemplarleaves a trace of its feature values in the weights connecting inputfeatures to the output unit, and the final pattern of generalization is

5 Figure 9b shows the median pattern of generalization over the threesuperordinate categories rather than the mean because the MAP generali-zations are always either 0 or 1, and thus the mean is sometimes notrepresentative of the model’s all-or-none predictions.

Figure 9. Predictions of two variants of the Bayesian model. (a) Without the size principle, Bayesiangeneralization behaves like an exemplar-similarity computation. (b) Without hypothesis averaging, Bayesiangeneralization follows an all-or-none, rulelike pattern. MAP Bayes ! maximum a posteriori Bayes approach;sub. ! subordinate; super. ! superordinate.






Results








Results








Results








Results




Full Bayesian model

Weak Bayesian model Most likely hypothesis only

P(X | h ) = [ 1height(h ) + ϵ

]n

P(X | h ) = 1

p(y ∈ C |X) = ∑h ∈H

P(y ∈ C | h )p(h |X)

generalization isn’t sharp enough generalization is too sharp

proportional to the average of the generalization (or “similarity”)gradients produced by each exemplar individually. Second, all ofthese models fail to converge to the appropriate level of specificitygiven multiple examples, because they all lack the size principlefor reweighting multiple consistent hypotheses to prefer the hy-pothesis most likely to have produced the observed examples.When the Hebbian learning network is presented with multipleexamples in the same subordinate category (e.g., three Dalma-tians), the correlation between the output unit and input featuresspecific to Dalmatians is no greater than the correlation betweenoutput and input features that apply to all dogs, or all animals,because every Dalmatian exemplar activates the dog and animalunits as well as the Dalmatian units. Because these correlations areindependent of the number of examples observed, the Hebbianmodel cannot explain why generalization beyond the subordinateconcept decreases as more examples lying strictly within thesubordinate are observed (e.g., why seeing three Dalmatian exem-plars leads to lower generalization to other dogs relative to seeingjust one Dalmatian exemplar). The same problem afflicts morepowerful associative learning mechanisms, such as standard neuralnetworks trained using back-propagation of errors (Rumelhart,Hinton, & Williams, 1986) and the recent associative models ofword learning (Colunga & Smith, 2005; Regier, 2003, 2005),which are also defined solely in terms of the statistics of input–output co-occurrence.

The Hebb rule, or other associative learning algorithms, couldbe modified to include some version of the size principle. Forinstance, we could allow learning rates to vary for different inputfeatures as a function of feature specificity. Such a move mightallow Bayesian and associative models of word learning to interactproductively. Our point here is not that connectionist models ofword learning are untenable but rather that generic associativelearning mechanisms based purely on correlations between observ-able features are not sufficient to explain how children or adultslearn the meanings of new words. More powerful mechanisms ofstatistical inference, such as our Bayesian framework, are neces-sary.

Figure 10d shows the predictions of a standard learning algo-rithm in the hypothesis elimination paradigm, known as the subsetprinciple (Berwick, 1986; Pinker, 1989; Siskind, 1996; see alsoWexler & Cullicover, 1980, for a discussion of subset-basedlearning in syntax acquisition, and Bruner et al., 1956, and Feld-man, 1997, for analogous proposals in category learning). Thesubset principle ranks all hypotheses by inclusion specificity:Hypothesis h i ranks higher than hypothesis h j if h j strictly includesh i—that is, if h j includes every object in h i as well as at least oneother object. A subset learner eliminates all hypotheses inconsis-tent with the observed examples and then generalizes in an all-or-none fashion according to the highest ranking remaining hypoth-esis—the most specific hypothesis consistent with the observedexamples. This approach is intuitively sensible and produces rea-sonable generalizations from multiple examples, but it is far tooconservative given just a single example.

The patterns of generalization from multiple examples under thesubset principle are essentially identical to those of MAP Bayes.Both approaches use just a single all-or-none hypothesis to guidegeneralization, chosen on the basis of inclusion specificity orposterior probability, respectively. These two criteria often con-verge because one component of posterior probability is likeli-hood, and under the size principle the likelihood is proportional tospecificity. The posterior probabilities of MAP Bayes also dependon the priors, which exert their strongest role when only oneexample has been observed. Here, the basic-level bias in the prioraccounts for why MAP Bayes generalizes differently than thesubset principle on one example—to all basic-level matches ratherthan to just the subordinate matches. The more examples havebeen observed, the stronger is the influence of the specificitypreference in the likelihood over the prior, and the more likely itis that MAP Bayes and the subset principle will coincide. In thelimit of infinite data, MAP Bayes (as well as our full Bayesianmodels) becomes equivalent to maximum likelihood, which underthe size principle is also equivalent to subset learning. Thus, thesubset principle can be justified as a rational statistical inferencewhen large numbers of examples have been observed, and it is

Figure 10. Predictions of four alternative, non-Bayesian models. Max ! maximum; Sim ! similarity; sub. !subordinate; super. ! superordinate.


What about an exemplar model?





Results








Results




Model Results














Results








Results








Results








Results




Full Bayesian model

Exemplar model

sim(y, C) = ∑x∈C

sim(y, x)





































Results








Results




Key point:

Exemplar model doesn’t care between 1 example and 3 examples

Conclusions

• Children can learn new concepts from just one or a few positive examples

• Children are sensitive to sampling assumptions, whether it’s three labels of the same exemplar or three different exemplars

• Bayesian models of concept learning provide an explicit model of sampling assumptions, and can learn concepts from just one or a few examples

• "Only a combination of sophisticated mental representations and sophisticated statistical inference machinery will be able to explain how adults and children can learn so many words so fast so accurately” (Xu & Tenenbaum)

Implication: neural network models and exemplar models (e.g., ALCOVE) are not up to the challenge

Developmental Science 10:3 (2007), pp 288–297 DOI: 10.1111/j.1467-7687.2007.00590.x

© 2007 The Authors. Journal compilation © 2007 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

Blackwell Publishing Ltd

BAYESIAN SPECIAL SECTION

Bayesian word learning

Sensitivity to sampling in Bayesian word learning

Fei Xu

1

and Joshua B. Tenenbaum

2

1. Department of Psychology, University of British Columbia, Canada2. Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, USA

Abstract

We report a new study testing our proposal that word learning may be best explained as an approximate form of Bayesianinference (Xu & Tenenbaum, in press). Children are capable of learning word meanings across a wide range of communicativecontexts. In different contexts, learners may encounter different sampling processes generating the examples of word–objectpairings they observe. An ideal Bayesian word learner could take into account these differences in the sampling process andadjust his/her inferences about word meaning accordingly. We tested how children and adults learned words for novel objectkinds in two sampling contexts, in which the objects to be labeled were sampled either by a knowledgeable teacher or by thelearners themselves. Both adults and children generalized more conservatively in the former context; that is, they restricted thelabel to just those objects most similar to the labeled examples when the exemplars were chosen by a knowledgeable teacher,but not when chosen by the learners themselves. We discuss how this result follows naturally from a Bayesian analysis, but notfrom other statistical approaches such as associative word-learning models.

Introduction

Models for how children learn the meanings of wordstraditionally fall into two classes. One class of modelstreats the process as inferential in nature, akin to reason-ing. Although the child presumably is not consciouslyworking out each step of the reasoning process and thecomputations may be done implicitly, the child learner isassumed to draw on a set of hypotheses about candidateword meanings and to evaluate these hypotheses basedon observed input using one or more principles of rationalinference (e.g. Bloom, 2000; Carey, 1978; Markman, 1989;Siskind, 1996). In contrast, associative models assumethat the learner represents a matrix of graded word–objectmappings, and the strengths of these mappings are incre-mentally increased or decreased over time given repeatedexposures (e.g. Colunga & Smith, 2005; Gasser & Smith,1998; Regier, 2003, 2005).

We will argue for an alternative view that combinesaspects of both approaches: the basic architecture isa form of rational hypothesis-driven inference, but theinferential logic is Bayesian and hence shows somethingof the graded statistical character of associative models(Tenenbaum & Xu, 2000; Xu & Tenenbaum 2005, in press).

Confronted with a novel word, the learner constructs ahypothesis space of candidate word meanings (i.e. lexi-calizable concepts) and a prior probability distributionover that hypothesis space. Given one or more examplesof objects labeled by the new word, the learner updates theprior to a posterior distribution of beliefs based on thelikelihood of observing these examples under each candi-date hypothesis. The prior represents any knowledge(due to previous learning or innate endowment) aboutwhich meanings are more or less likely to be the targetof the new word, independent of the observed examples.The likelihood is based on the sampling process pre-sumed to have generated the observed object–label pairs.

Recent studies of word learning with adults andchildren provide some initial evidence for this account.These studies test generalization: participants are shownone or more examples of a novel word (e.g. ‘blicket’) andare asked to judge which objects from a test set the wordalso applies to. Xu and Tenenbaum (in press) demon-strated that in learning object kind labels at differentlevels of the hierarchy (i.e. subordinate, basic level, andsuperordinate), both the generalization patterns of adultsand 4-year-old children were sensitive to the number andthe span of the examples, in the ways predicted by a

Address for correspondence: F. Xu, Department of Psychology, 2136 West Mall, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada;e-mail: [email protected] or J.B. Tenenbaum, Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, 77 MassachusettsAvenue, Cambridge, MA 02139, USA; e-mail: [email protected]

Bayesian word learning 291

© 2007 The Authors. Journal compilation © 2007 Blackwell Publishing Ltd.

is shown a positive example, the label is generated first(depending on what the speaker wants to communicatein a particular context), and then an object is chosenfrom the set of stimuli with that label. This causal orderis reversed in weak sampling: the learner first picks anobject to be labeled out of the total set of stimuli, andthen the teacher responds with either a positive ornegative label depending on whether the object chosenfalls inside the word’s positive extension.

This difference in causal order affects the inferencesthat the learner is licensed to draw under strong or weaksampling. Under strong sampling, the fact that a particularexample is positive is not informative about the meaningof the word – only that the example is being generatedby a knowledgeable speaker who either intends to pro-

vide positive examples in an ostensive teaching context,or is merely using words correctly (to refer to theirpositive instances) in the course of natural conversationwith other competent speakers. However, the particularobject chosen for labeling is informative about the word’smeaning, because it is presumed to be a random samplefrom the word’s positive extension. These dependenciesare reversed under weak sampling. The choice of whichobject to label does not depend directly on the word’smeaning, because the learner is choosing the examplesand does not know the word’s meaning. However, thelabel observed does depend on the meaning, because thelabels are provided by a competent user who knows whatthe word means and will presumably label the learner’schosen object positively or negatively according to whetheror not it falls in the word’s extension. The differences inthese patterns of causal dependencies, and the probabilisticdependencies they imply, lead a Bayesian learner tomake qualitatively different kinds of inferences in theselearning situations.

More formally, in the strong sampling condition, wecan write

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(4)

This likelihood function reflects what we have called thesize principle (Tenenbaum, 1999; Tenenbaum & Griffiths,2001): more specific meanings, with smaller extensions,are more likely than more general meanings, with largerextensions, when both are consistent with a given set ofexamples; and the preference for more specific meanings

Figure 1 (a) A schematic illustration of the hypothesis space used to model generalization in the experiment, for the stimuli shown in (b). (b) One set of stimuli used in the experiment, as they were shown to participants.

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Yes, that’s right.That’s right too



each subordinate category, and 15 objects in each basic-level category. Four nonsense words were used (‘blicket’,‘tupa’, ‘wug’, and ‘fep’). Which word was used to referto which picture and which picture was shown first werecounterbalanced across participants.

Teacher-driven condition

With child participants, the experimenter began by ask-ing the child to play a game with her. She pointed to anobject in the picture and said to the child, ‘See this? It’sa blicket.’ She then pointed to two other objects fromthe same subordinate category, one at a time, and saidto the child, ‘See this one? It’s a blicket.’ She then askedthe child to choose a sticker for paying attention anddoing a good job in the game. The experimenter thenproceeded to the test phase, in which the child was askedto decide which other objects were also blickets. Theexperimenter pointed to a total of five other objects, andfor each one she asked the child, ‘Is this a blicket?’ Thefive test objects were, in order of questioning, a sub-ordinate match, a non-match from the other basic-levelcategory, a basic-level match from a different subordinate,another subordinate match, and another basic-level match.Participants received no feedback on their answers totest questions. The experimenter then presented the childwith the second set of objects and went through thesame procedure, using a different novel word. At the endof the study, each child was allowed to choose anothersticker for doing a good job.

The procedure for adults was identical to that for chil-dren. The adults were told that the study was initiallydesigned for preschoolers, and that stickers were givenout to keep the child on task.

Learner-driven condition

The procedure was identical to that of the teacher-drivencondition with the following critical difference. Afterpresenting the first example with the phrase, ‘See this?It’s a blicket.’ The experimenter then asked the child,‘Can you point to two other blickets? If you get both ofthem right, you get a sticker!’ Once the child picked twomore objects, the experimenter confirmed that the childhad correctly found two blickets (regardless of the child’sselections) and the child was allowed to choose a stickeras a reward. The experimenter then proceeded to the testphase, pointing to five test objects and asking for eachone, ‘Is this a blicket?’ just as in the teacher-driven con-dition. In both conditions, these five test questions wereasked only after participants had seen three labeledexamples; the only difference between conditions was theprocess by which the second and third labeled examples

were sampled. The procedure for adults was again identicalto that for children. With the exception of one child(whose data were excluded from the analyses), all par-ticipants pointed to two other objects from the samesubordinate category when asked to point to two otherblickets.

Results

Figure 2 summarizes responses to the test questions inboth conditions, in terms of the frequencies with whichparticipants generalized to different levels of the cate-gory hierarchy. Over the five test trials, no participantgeneralized a new word to any object from the otherbasic-level category, and every participant responded ina way that was consistent with a preferred level of gen-eralization for a given word. That is, for each novelword, each participant either generalized to just the twosubordinate-level matches, or to those objects and thetwo basic-level matches. We calculated the percentagesof ‘yes’ responses for both the subordinate and thebasic-level matches. In the teacher-driven condition,children generalized the novel word at the subordinate

Figure 2 Percentages of generalization responses at the subordinate and basic levels, for adults and children in both teacher-driven (a) and learner-driven (b) conditions.Corresponding posterior probabilities for subordinate and basic-level hypotheses are shown for the Bayesian model.










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Key point: exemplar models, ALCOVE, and neural networks don’t differentiate based on sampling process

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Cognitive Science 32 (2008) 108–154Copyright C⃝ 2008 Cognitive Science Society, Inc. All rights reserved.ISSN: 0364-0213 print / 1551-6709 onlineDOI: 10.1080/03640210701802071

A Rational Analysis of Rule-Based Concept Learning

Noah D. Goodmana, Joshua B. Tenenbauma, Jacob Feldmanb,Thomas L. Griffithsc

aDepartment of Brain and Cognitive Sciences, Massachusetts Institute of TechnologybDepartment of Psychology, Rutgers University

cDepartment of Psychology, University of California, Berkeley

Abstract

This article proposes a new model of human concept learning that provides a rational analysisof learning feature-based concepts. This model is built upon Bayesian inference for a grammaticallystructured hypothesis space—a concept language of logical rules. This article compares the modelpredictions to human generalization judgments in several well-known category learning experiments,and finds good agreement for both average and individual participant generalizations. This articlefurther investigates judgments for a broad set of 7-feature concepts—a more natural setting in severalways—and again finds that the model explains human performance.

Keywords: Concept learning; Categorization; Bayesian induction; Probabilistic grammar; Rules;Language of thought

But what are concepts save formulations and creations of thought, which, instead of giving us thetrue form of objects, show us rather the forms of thought itself? (Cassirer, 1946, p. 7)

The study of concepts—what they are, how they are used and how they are acquired—hasprovided one of the most enduring and compelling windows into the structure of the humanmind. What we look for in a theory of concepts, and what kinds of concepts we look at,depends on the functions of concepts that interest us. Three intuitions weave throughout thecognitive science literature (e.g., see Fodor, 1998; Murphy, 2002):

1. Concepts are mental representations that are used to discriminate between objects, events,relationships, or other states of affairs. Cognitive psychologists have paid particular atten-tion to concepts that identify kinds of things—those that classify or categorize objects—andsuch concepts are our focus here. It is clear how an ability to separate objects accordingto kind could be critical to survival. To take a classic example, a decision about whether

Correspondence should be addressed to Noah D. Goodman, Massachusetts Institute of Technology, 77 Mas-sachusetts Ave., MIT Building 46-4053, Cambridge, MA 02139. E-mail: [email protected]

Motivating rational rules

• The author’s lay out three themes in concept learning• Concepts are used to discriminate between objects, events,

relationships, etc.• Concepts are learned inductively from sparse and noisy data• Concepts are compositional and are formed by combining simpler

concepts

Rational rules is an attempt to combine these themes into a single model by combining the “classical view”/rule learning with probabilistic inference

Key question: The classical view (e.g., rules and definitions) does not account for graded effects in categorization and learning. Can we account for these effects by performing Bayesian inference over rules and definitions?

A B

color

textureslash

A B

Type I Type II

Model’s aim is to learn defining rules for concepts

130 N. D. Goodman, J. B. Tenenbaum, J. Feldman, T. L. Griffiths/Cognitive Science 32 (2008)

Table 5The six concepts with three features, four positive and four negative examples, studied first in Shepard et al. (1961).

I II III IV V VI

+ 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0+ 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 − 0 0 1+ 0 1 0 − 0 1 0 + 0 1 0 + 0 1 0 + 0 1 0 − 0 1 0+ 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 + 0 1 1− 1 0 0 − 1 0 0 − 1 0 0 + 1 0 0 − 1 0 0 − 1 0 0− 1 0 1 − 1 0 1 + 1 0 1 − 1 0 1 − 1 0 1 + 1 0 1− 1 1 0 + 1 1 0 − 1 1 0 − 1 1 0 − 1 1 0 + 1 1 0− 1 1 1 + 1 1 1 − 1 1 1 − 1 1 1 + 1 1 1 − 1 1 1

membership, and the remaining features are uninformative. For Concept II the first two featuresare informative; for example, the complexity 4 formula:

((f1(x) = 1) ∧ (f2(x) = 1)) ∨ ((f1(x) = 0) ∧ (f2(x) = 0))

is the simplest perfect rule for this concept. In contrast, all three features are informative forConcepts III, IV, V, and VI. Concept III admits the relatively simple formula

((f1(x) = 0) ∧ (f3(x) = 0)) ∨ ((f2(x) = 0) ∧ (f3(x) = 1)),

whereas Concepts IV, V, and VI do not admit any perfect rules of low complexity. However,IV, and V both admit imperfect, but useful, rules of low complexity, whereas VI has no usefulsimple rules at all.

A well-replicated finding concerning human errors (Shepard et al., 1961) is that theseconcepts vary reliably in difficulty, reflecting the above complexity and informativeness con-siderations: I < II < III = IV = V < VI (ordered from least to most difficulty, where “=”indicates no reliable difference in diffculty). The RRDNF model predicts these qualitative find-ings: error rates (via posterior probability, when b= 3) of 0%, 17%, 24%, 24%, 25%, 48%for concepts I, II, III, IV, V, and VI, respectively. This ordering is predicted for a fairly widerange of parameter values, although an inversion is predicted at b= 1: concept II is then moredifficult than concepts III, IV, and V. It is intriguing that this inversion has been experimentallyobserved in humans when the stimulus dimensions are nonseparable (Nosofsky & Palmeri,1996), but further work will be needed to determine whether this is an illuminating or acci-dental prediction of the model. (This inversion, which is also seen in rhesus monkeys [Smith,Minda, & Washburn, 2004], is predicted by the ALCOVE model when attention learning isdisabled.)

However, people are not bound to attend to the smallest set of informative features—indeed,selective attention is particularly interesting in light of the implied trade-off between accuracyand number of features attended. Medin, Altom, Edelson, and Freko (1982) demonstratedthis balance by studying the category structure shown in Table 6. This structure affords twostrategies: Each of the first two features are individually diagnostic of category member-ship, but not perfectly so, whereas the correlation between the third and fourth features is

N. D. Goodman, J. B. Tenenbaum, J. Feldman, T. L. Griffiths/Cognitive Science 32 (2008) 129

1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Complexity

Post

erio

r pr

obab

ility

Concept LSConcept NLS

Fig. 6. Posterior complexity distribution of the RRDNF model (b= 3) on the two category structures from Medinand Schwanenflugel (1981; see Table 4). The model shows greater dependence on simple rules for Concept LS(linearly separable) than Concept NLS (not linearly separable).

complexity and feature weights) is a measure of selective attention exhibited by the model:The posterior complexity weights describe the extent to which the model favors simplerformulae (which will have fewer features), whereas the posterior feature weights directlydescribe the informativeness of each feature, as estimated by the model. It has been notedbefore (Navarro, 2006) that selective attention effects emerge naturally from the Bayesianframework. In our setting selective attention can be understood as the effect of updating theuncertainty over production probabilities as evidence accumulates. Indeed, as the prior overτ—initially uniform—is updated, it will often concentrate, becoming tightly peaked on a sub-set of productions. For instance, if only the first of three features is informative, the posteriordistribution on production P → F1 will become larger, whereas the posteriors on P → F2and P → F3 will be small (and these values will be reflected in the posterior feature weights;see Appendix D). The changing importance of the productions τ have been marginalizedaway in the summary prior, Equation 4, but the effects will still be felt in model predic-tions. As a result the inferences of the Rational Rules model will depend most sensitively onthe informative features—this is the manner in which Bayesian models implement selectiveattention.

Shepard et al. (1961), in one of the first studies to demonstrate selective attention effects,compared difficulty in learning the six concepts in Table 5 (these are the six concepts withthree Boolean features, four positive and four negative examples). These concepts differ inthe number of dimensions that must be attended to, in the complexity of their simplest perfectrule, and in the number of imperfect, but useful, simple rules. To learn Concept I it is onlynecessary to consider the first feature, that is, the rule (f1(x) = 0) perfectly predicts category

f1(x) : colorf2(x) : slashf3(x) : texture

(color is purple) (color is blue AND has slash) OR (color is purple AND has no slash)

color

textureslash

A B

Type III


(color is blue AND has no texture) OR (has slash and has texture)



I II III IV V VI

+ 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0+ 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 − 0 0 1+ 0 1 0 − 0 1 0 + 0 1 0 + 0 1 0 + 0 1 0 − 0 1 0+ 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 + 0 1 1− 1 0 0 − 1 0 0 − 1 0 0 + 1 0 0 − 1 0 0 − 1 0 0− 1 0 1 − 1 0 1 + 1 0 1 − 1 0 1 − 1 0 1 + 1 0 1− 1 1 0 + 1 1 0 − 1 1 0 − 1 1 0 − 1 1 0 + 1 1 0− 1 1 1 + 1 1 1 − 1 1 1 − 1 1 1 + 1 1 1 − 1 1 1


((f1(x) = 1) ∧ (f2(x) = 1)) ∨ ((f1(x) = 0) ∧ (f2(x) = 0))


((f1(x) = 0) ∧ (f3(x) = 0)) ∨ ((f2(x) = 0) ∧ (f3(x) = 1)),




Model’s aim is to learn defining rules for concepts

Rational rules model

p(F | l(X); X) ∝ p(l(X) |F; X)p(F)

F : formula that defines a concept (Rule)

X : set of observed examples


Likelihood (noisy labeling according to formula)

Prior Based on derivation of formula under a probability context free grammar (favors short formulas and re-use of derivational steps)

l(X) : labels provided to examples

Generalizing to a new example y

p(y ∈ l | l(X); X) = ∑F

P(y ∈ C |F)p(F | l(X); X)

p(l(X) |F; X) ∝ e−bQl(F)

Ql(F) number of labels that have been misaligned according to formula

p(F)


(S1) S → ∀x ℓ(x)⇔(D)

(D1) D → (C) ∨D

(D2) D → False

(C1) C → P ∧C

(C2) C → True

(P1) P → F1

...

(PN) P → FN

(F11) F1 → f1(x) = 1

(F12) F1 → f1(x) = 0...

(FN1) FN → fN (x) = 1

(FN2) FN → fN (x) = 0

Fig. 1. Production rules of the DNF grammar. Note S is the start symbol, and D,C, P, Fi the other non-terminals.Productions (Fi1) and (Fi2) can be naturally extended to “decision boundary” predicates—for example, F1 →f1(x ) < 2.

F12 :

∀x (x)⇔(((f1(x)=0 ) ∧True) ∨False)

∀x (x)⇔((F1 ∧True) ∨False)

∀x (x)⇔((P ∧True) ∨False)

∀x (x)⇔((P ∧C) ∨False)

∀x (x)⇔((C) ∨False)

∀x (x)⇔((C) ∨D)

∀x (x)⇔(D)

SS1 :

D1 :

D2 :

C1 :

C2 :

P1 :

Fig. 2. Derivation of a formula from the DNF grammar.


(S1) S → ∀x ℓ(x)⇔(D)

(D1) D → (C) ∨D

(D2) D → False

(C1) C → P ∧C

(C2) C → True

(P1) P → F1

...

(PN) P → FN

(F11) F1 → f1(x) = 1

(F12) F1 → f1(x) = 0...

(FN1) FN → fN (x) = 1

(FN2) FN → fN (x) = 0

Fig. 1. Production rules of the DNF grammar. Note S is the start symbol, and D,C, P, Fi the other non-terminals.Productions (Fi1) and (Fi2) can be naturally extended to “decision boundary” predicates—for example, F1 →f1(x ) < 2.

F12 :

∀x (x)⇔(((f1(x)=0 ) ∧True) ∨False)

∀x (x)⇔((F1 ∧True) ∨False)

∀x (x)⇔((P ∧True) ∨False)

∀x (x)⇔((P ∧C) ∨False)

∀x (x)⇔((C) ∨False)

∀x (x)⇔((C) ∨D)

∀x (x)⇔(D)

SS1 :

D1 :

D2 :

C1 :

C2 :

P1 :

Fig. 2. Derivation of a formula from the DNF grammar.

A grammar of rule-based concepts Derivation of rules:

Deriving a formula

A B

color

textureslash

A B

Type I Type II

Accounts for SHJ due to boolean complexity



I II III IV V VI

+ 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0+ 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 − 0 0 1+ 0 1 0 − 0 1 0 + 0 1 0 + 0 1 0 + 0 1 0 − 0 1 0+ 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 + 0 1 1− 1 0 0 − 1 0 0 − 1 0 0 + 1 0 0 − 1 0 0 − 1 0 0− 1 0 1 − 1 0 1 + 1 0 1 − 1 0 1 − 1 0 1 + 1 0 1− 1 1 0 + 1 1 0 − 1 1 0 − 1 1 0 − 1 1 0 + 1 1 0− 1 1 1 + 1 1 1 − 1 1 1 − 1 1 1 + 1 1 1 − 1 1 1


((f1(x) = 1) ∧ (f2(x) = 1)) ∨ ((f1(x) = 0) ∧ (f2(x) = 0))


((f1(x) = 0) ∧ (f3(x) = 0)) ∨ ((f2(x) = 0) ∧ (f3(x) = 1)),




N. D. Goodman, J. B. Tenenbaum, J. Feldman, T. L. Griffiths/Cognitive Science 32 (2008) 129

1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Complexity

Post

erio

r pr

obab

ility


Fig. 6. Posterior complexity distribution of the RRDNF model (b= 3) on the two category structures from Medinand Schwanenflugel (1981; see Table 4). The model shows greater dependence on simple rules for Concept LS(linearly separable) than Concept NLS (not linearly separable).

complexity and feature weights) is a measure of selective attention exhibited by the model:The posterior complexity weights describe the extent to which the model favors simplerformulae (which will have fewer features), whereas the posterior feature weights directlydescribe the informativeness of each feature, as estimated by the model. It has been notedbefore (Navarro, 2006) that selective attention effects emerge naturally from the Bayesianframework. In our setting selective attention can be understood as the effect of updating theuncertainty over production probabilities as evidence accumulates. Indeed, as the prior overτ—initially uniform—is updated, it will often concentrate, becoming tightly peaked on a sub-set of productions. For instance, if only the first of three features is informative, the posteriordistribution on production P → F1 will become larger, whereas the posteriors on P → F2and P → F3 will be small (and these values will be reflected in the posterior feature weights;see Appendix D). The changing importance of the productions τ have been marginalizedaway in the summary prior, Equation 4, but the effects will still be felt in model predic-tions. As a result the inferences of the Rational Rules model will depend most sensitively onthe informative features—this is the manner in which Bayesian models implement selectiveattention.

Shepard et al. (1961), in one of the first studies to demonstrate selective attention effects,compared difficulty in learning the six concepts in Table 5 (these are the six concepts withthree Boolean features, four positive and four negative examples). These concepts differ inthe number of dimensions that must be attended to, in the complexity of their simplest perfectrule, and in the number of imperfect, but useful, simple rules. To learn Concept I it is onlynecessary to consider the first feature, that is, the rule (f1(x) = 0) perfectly predicts category


(color is purple) (color is blue AND has slash) OR (color is purple AND has no slash)

color

textureslash

A B

Type III


(color is blue AND has slash) OR (has slash and has texture)



I II III IV V VI

+ 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0+ 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 + 0 0 1 − 0 0 1+ 0 1 0 − 0 1 0 + 0 1 0 + 0 1 0 + 0 1 0 − 0 1 0+ 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 − 0 1 1 + 0 1 1− 1 0 0 − 1 0 0 − 1 0 0 + 1 0 0 − 1 0 0 − 1 0 0− 1 0 1 − 1 0 1 + 1 0 1 − 1 0 1 − 1 0 1 + 1 0 1− 1 1 0 + 1 1 0 − 1 1 0 − 1 1 0 − 1 1 0 + 1 1 0− 1 1 1 + 1 1 1 − 1 1 1 − 1 1 1 + 1 1 1 − 1 1 1


((f1(x) = 1) ∧ (f2(x) = 1)) ∨ ((f1(x) = 0) ∧ (f2(x) = 0))


((f1(x) = 0) ∧ (f3(x) = 0)) ∨ ((f2(x) = 0) ∧ (f3(x) = 1)),




Accounts for SHJ due to boolean complexity

Medin & Schaffer Experiment 2famous “5-4” category structure

Key comparison is stimulus 4 vs stimulus 7• Prototype model would predict stimulus 4 is easier to learn

It’s more similar to the prototype• Exemplar model would predict stimulus 7 is easier to learn

Has two near neighbors, 15 and 4• Behavior results favor exemplar model: stimulus 7 had

fewer error (FE) and higher confidence rating

222 DOUGLAS L. MEDIN AND MARGUERITE M. SCHAFFER

"A" STIMULI

STIMULUSNUMBER DIMENSION VALUES

C F S N

47

1513

5

1 1 11 0 11 0 11 1 0O i l

STIMULUSNUMBER

13689

1116

00111

43344

TRAINING STIMULI"R" STTMIII i

Et,9,3,2,8.5

NEW

STIMULUSRAT_ NUMBER DIMENSION VALUESING C F S N FE

4 , 8 1 2 1 1 0 0 5 , 55 . 4 2 0 1 1 0 5 , 25 , 1 1 4 0 0 0 1 3 , 95 , 2 1 0 0 0 0 0 3 , 15,2

TRANSFER STIMULI

RAT-

5,05,15,25.5

DIMENSION VALUES RATINGC F S N A-PREDICTED E-PREDICTED

1110000

0010101

0011010

1 3.70 4,41 5,30 4,11 3,31 4.10 4,9

Figure 4. Design of Experiment 2. (C, P, S, and N refer to the dimensions of color, form, size, andnumber, respectively. The stimulus numbers are carried over from Experiment 1. Fe refers to averageerrors during learning. Rating scores may vary from 1 to 6, with 3.5 representing chance [nohdifferential]classification.)

eter values that would alter this predictionwould place numerous other testable con-straints on the data.

Method

Subjects. Thirty-two volunteers were solicitedthrough ads in local newspapers. The subjects, men andwomen ranging in ages from 17 to 30 years, were paid$2.50 for the experimental session. The subjects had notparticipated in the first experiment.

Stimuli, Sixteen stimulus cards with geometricforms drawn on them were used. Nine cards were usedin training and seven additional cards were used intransfer. The geometric forms were like those from thepreceding experiment, except that the dimension ofnumber was substituted for the dimension of position.The number dimension was represented by either asingle geometric form centered on the card or by twogeometric forms each centered on their respectivehalves of the card.

The assignment of abstract notation to individualstimulus cards varied from subject to subject exactlyas in the first experiment. That is, the assignment of

stimulus cards to conditions and category labels wasexactly counterbalanced.

Procedure. The procedure followed that used inthe first part of Experiment 1: initial training, followedby a S-10-minute interpolated activity, followed by atransfer task involving both training and new transferstimuli.

The instructions for training were those used inExperiment 1. Training consisted of up to 16 runsthrough the list of 9 training stimuli with a learningcriterion of 1 errorless run. Other procedural detailsfollowed those of Experiment 1, including the inter-polated activity and the transfer test instructions andprocedure.

Results

Learning. The learning task was of moder-ate difficulty; 19 of the 32 subjects learned theclassification task within the maximum limitof 16 runs. Overall, subjects averaged 18%errors on the last run through the list, butvirtually all subjects showed some improve-

Prototype: 1 1 1 1 Prototype: 0 0 0 0


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

A1A2

A3

A4

A5

B1

B2

B3B4

T1

T2

T3

T4T5

T6

T7

RR predictions

Hum

an m

ean

Fig. 3. Comparison of human judgments with RRDNF model predictions: mean probability of category A judgmentsafter training on the category structure of Medin and Schaffer (1978). See Table 3 for human and RRDNF model(b= 1). The fit between model and human data is R2 = 0.98.

enhancement: most high probability formulae agree on the classification of B4, whereas feweragree on the classifications of the other training examples. However, note that the typicalitygradient of training examples is not entirely determined by similarity to the prototypes. Forinstance, training example A2 and A1 are equally typical of category A and more typical thanexamples A4 and A5 (according to both humans and the Rational Rules model), however A1,

)b()a(

1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Complexity

Post

erio

r co

mpl

exity

wei

ght

1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Feature

Post

erio

r fe

atur

e w

eigh

t

Fig. 4. (a) Posterior complexity distribution of the RRDNF model (b= 1) for the category structure of Medinand Schaffer (1978; see Table 3). (b) Posterior feature weights for this category structure. Together these weightdistributions indicate that the RRDNF model focuses on simple rules along features 1 and 3.

Medin & Schaffer Experiment 2Rational rules can account for enhancement by neighboring exemplarsN. D. Goodman, J. B. Tenenbaum, J. Feldman, T. L. Griffiths/Cognitive Science 32 (2008) 125

Table 3The category structure of Medin & Schaffer (1978), with the human data of Nosofsky et al. (1994a), and thepredictions of the Rational Rules model (b = 1).

Object Feature Values Human RRDNF

A1 0001 0.77 0.82A2 0101 0.78 0.81A3 0100 0.83 0.92A4 0010 0.64 0.61A5 1000 0.61 0.61B1 0011 0.39 0.47B2 1001 0.41 0.47B3 1110 0.21 0.21B4 1111 0.15 0.07T1 0110 0.56 0.57T2 0111 0.41 0.44T3 0000 0.82 0.95T4 1101 0.40 0.44T5 1010 0.32 0.28T6 1100 0.53 0.57T7 1011 0.20 0.13

The best (highest posterior probability) formulae according to the Rational Rules model areshown in Table 2. For small values of b (those that are more permissive of outliers), the bestformulae are single dimension rules. Note that the posterior probabilities of these formulaereflect their predictiveness on the training examples, and in particular that formula f4(x) = 0,although not the best, has significantly greater posterior probability than formula f2(x) = 0.In Fig. 4 we have plotted the posterior complexity weights and the posterior feature weights ofthe Rational Rules model for b = 1. We see that this pattern is maintained when consideringthe entire posterior: most of the weight is on simple formulae along Features 1 and 3, followedby Feature 4, then Feature 2.

The object T3 = 0000 is the prototype of category A, in the sense that most of the examples ofcategory A are similar to this object (differ in only one feature), whereas most of the examplesof category B are dissimilar. Although it never occurs in the training set, the importance of thisprototype is reflected in the human transfer judgments (Table 3 and Fig. 3): T3 is, by far, themost likely transfer object to be classified as category A. The Rational Rules model predictsthis “prototype enhancement effect” (Posner & Keele, 1968). This prediction results becausethe highest posterior probability formulae (Table 2) all agree on the categorization of T3, socombine (together with lower probability formulae) to enhance the probability that T3 is incategory A.

The degree of typicality, or recognition rate for training examples, is often taken as a usefulproxy for category centrality (Mervis & Rosch, 1981) because it correlates with many of thesame experimental measures (such as reaction time). In Table 3 and Fig. 3, we see greatertypicality for the prototype of category B, the object B4 = 1111, than for other trainingexamples: although presented equally often it is classed into category B far more often. TheRational Rules model also predicts this typicality effect, in a manner similar to prototype

more like prototype (all 0’s)more like other exemplars

probability of classifying as category “A”

R2 = 0.98

Rational rules accounts for this by reliance on rules that often use Features 1 and 3, and not often Feature 2


Maddox & Ashby, 1993) suggests that feature predicates capturing general linear or quadraticdecision boundaries may be appropriate in some situations.

4.4. Individual generalization patterns

Nosofsky et al. (1994) investigated the pattern of generalizations made by individual par-ticipants, that is, they reported the proportion of participants giving each set of answers to thegeneralization questions. One may wonder whether it is necessary to consider these general-ization patterns in addition to group averages for each question. As noted in Nosofsky andPalmeri (1998), even the best binomial model does very poorly at predicting individual gener-alization patterns (R2 = 0.24 in the case of Nosofsky et al., 1994, Experiment 1), although, byconstruction, it perfectly predicts the group average for each generalization question. There-fore the pattern of generalizations provides an additional, more fine grained, probe for testingconcept learning models.

To understand how the Rational Rules model can predict these generalization patterns,recall the hypothesis sampling assumption discussed earlier: each individual has a singlehypothesis that is drawn from the posterior over formulae. The pattern of judgments made byeach individual is then determined by this hypothesis, with additional response noise η. If weassume a single value of the parameters (band η) for all participants, the best fit of the RRDNFmodel explains R2 = 0.85 of the variance in human generalization for the 36 generalizationpatterns reported in (Nosofsky et al., 1994). The RULEX model also does well, R2 = 0.86, butuses several more free parameters. As with RULEX, the qualitative match of the Rational Rulesmodel to human judgments is good (see Fig. 10). Also as with RULEX, the generalizationpattern ABABBAB is underpredicted by the model (cf. Johansen & Palmeri, 2002).

0

0.05

0.1

0.15

0.2

AA

AA

AA

AA

AA

AB

AB

AA

AB

AA

BA

AA

BA

BB

AA

AB

BA

BA

AA

BB

BA

AA

AB

BB

BA

AB

BA

BA

AA

BB

BB

BA

BA

AA

AB

AB

AA

BA

BA

BA

BA

AA

AB

AB

AA

BA

BA

BA

BA

AB

AB

AB

BA

BA

BB

AB

AB

AB

BB

BA

BB

AB

AB

AB

BB

AA

BA

BB

BB

BA

BA

AA

BA

BB

AA

BB

AB

BA

AB

BB

BB

AB

BB

BB

BB

AA

AA

AB

BA

AA

AB

BB

AA

BA

BB

BA

AB

BB

BB

AB

AA

AB

BA

BA

AB

BB

AB

AB

AB

BA

BA

BB

BB

AB

BA

BB

BA

BB

BA

BB

AB

BB

BB

BB

AA

BB

ModelHuman

Fig. 10. Individual generalization patterns: the probability of responding with the indicated categorizations forthe seven transfer stimuli of Table 3. Human data from Nosofsky, Palmeri, and McKinley (1994), Experiment 1.The model values are for parameters b= 4, η = 0.09. Agreement of model with human data is good: R2 = 0.85,rmsd = 0.016.

Individual generalization patterns222 DOUGLAS L. MEDIN AND MARGUERITE M. SCHAFFER

"A" STIMULI

STIMULUSNUMBER DIMENSION VALUES

C F S N

47

1513

5

1 1 11 0 11 0 11 1 0O i l

STIMULUSNUMBER

13689

1116

00111

43344

TRAINING STIMULI"R" STTMIII i

Et,9,3,2,8.5

NEW

STIMULUSRAT_ NUMBER DIMENSION VALUESING C F S N FE

4 , 8 1 2 1 1 0 0 5 , 55 . 4 2 0 1 1 0 5 , 25 , 1 1 4 0 0 0 1 3 , 95 , 2 1 0 0 0 0 0 3 , 15,2

TRANSFER STIMULI

RAT-

5,05,15,25.5

DIMENSION VALUES RATINGC F S N A-PREDICTED E-PREDICTED

1110000

0010101

0011010

1 3.70 4,41 5,30 4,11 3,31 4.10 4,9

Figure 4. Design of Experiment 2. (C, P, S, and N refer to the dimensions of color, form, size, andnumber, respectively. The stimulus numbers are carried over from Experiment 1. Fe refers to averageerrors during learning. Rating scores may vary from 1 to 6, with 3.5 representing chance [nohdifferential]classification.)

eter values that would alter this predictionwould place numerous other testable con-straints on the data.

Method

Subjects. Thirty-two volunteers were solicitedthrough ads in local newspapers. The subjects, men andwomen ranging in ages from 17 to 30 years, were paid$2.50 for the experimental session. The subjects had notparticipated in the first experiment.

Stimuli, Sixteen stimulus cards with geometricforms drawn on them were used. Nine cards were usedin training and seven additional cards were used intransfer. The geometric forms were like those from thepreceding experiment, except that the dimension ofnumber was substituted for the dimension of position.The number dimension was represented by either asingle geometric form centered on the card or by twogeometric forms each centered on their respectivehalves of the card.

The assignment of abstract notation to individualstimulus cards varied from subject to subject exactlyas in the first experiment. That is, the assignment of

stimulus cards to conditions and category labels wasexactly counterbalanced.

Procedure. The procedure followed that used inthe first part of Experiment 1: initial training, followedby a S-10-minute interpolated activity, followed by atransfer task involving both training and new transferstimuli.

The instructions for training were those used inExperiment 1. Training consisted of up to 16 runsthrough the list of 9 training stimuli with a learningcriterion of 1 errorless run. Other procedural detailsfollowed those of Experiment 1, including the inter-polated activity and the transfer test instructions andprocedure.

Results

Learning. The learning task was of moder-ate difficulty; 19 of the 32 subjects learned theclassification task within the maximum limitof 16 runs. Overall, subjects averaged 18%errors on the last run through the list, butvirtually all subjects showed some improve-

famous “5-4” category structure

Probability of responding with the indicated categorizations of 7 transfer stimuli

People don’t necessarily favor linearly separable categories (as accounted for by ALCOVE)

linearly separable condition: people had 39.5 errors on average

non-linearly separable condition: people had 38.0 errors on average

(criterion was two error-free passes through stimuli)


)b()a(

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Err

or r

ate

Block


0 1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Err

or p

roba

bilit

y

b


Fig. 5. (a) The human data from Medin and Schwanenflugel ( 1981) for the category structures in Table 4, showingthat linearly separable Concept LS was more difficult to learn than Concept NLS, which is not linearly separable.(b) Predictions of the RRDNF model: the probability of an incorrect response versus the outlier parameter b.

agreement with the human data, predicting more errors on the linearly separable concept (andnote that no parameters were fit in these model results). The RULEX model also predicts therelative diculty of Concept LS over Concept NLS (Nosofsky et al., 1994), which suggests thatthis is a rather general prediction of rule-based models.

To understand this result, note that, although the two concepts support equally informativecomplexity 1 rules (that is single-feature strategies), Concept NLS supports more informativerules of complexity 2, 3, and 4 than does Concept LS. For example, the complexity 4 formula(f1(x) = 0 ∧f2(x) = 0) ∨ (f3(x) = 0 ∧f4(x) = 0) discriminates perfectly for Concept NLS,whereas there is no complexity 4 formula that does so for Concept LS. The RRDNF modelrelies more heavily on these rules of complexity 2, 3, and 4 for Concept NLS than for ConceptLS, see the plots of posterior complexity in Fig. 6, which results in a difference in accuracy.The model does not, however, simply use the most informative rules (after all there are alwaysperfectly predictive rules of very high complexity), but balances predictive accuracy againstsimplicity—it places weight on highly informative and moderately complex rules for ConceptNLS, but, finding no such rules for Concept LS, places the majority of the weight on verysimple rules.

4.2. Selective attention effects

It is important to be able to ignore uninformative features in a world, such as ours, inwhich every object has many features, any of which may be useful for classification. Thismotivates the long standing interest in selective attention in human concept learning (Kr-uschke, 1992): the tendency to consider as few features as possible to achieve acceptableclassification accuracy. We have seen a simple case of this already predicted by the RationalRules model: Single feature concepts were preferred to more complex concepts for the 5–4category structure (Fig. 4a). Indeed, each of the descriptive measures described earlier (the

Rational rules can learn NLS concepts more easily than LS

A complexity 4 rule perfectly discriminates the NLS case, but you need a more complex rule for the LS case

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