{"title": "Bayesian Modeling of Human Concept Learning", "book": "Advances in Neural Information Processing Systems", "page_first": 59, "page_last": 68, "abstract": null, "full_text": "Bayesian modeling of human concept learning \n\nJoshua B. Tenenbaum \n\nDepartment of Brain and Cognitive Sciences \n\nMassachusetts Institute of Technology, Cambridge, MA 02139 \n\njbt@psyche.mit.edu \n\nAbstract \n\nI consider the problem of learning concepts from small numbers of pos(cid:173)\nitive examples, a feat which humans perform routinely but which com(cid:173)\nputers are rarely capable of. Bridging machine learning and cognitive \nscience perspectives, I present both theoretical analysis and an empirical \nstudy with human subjects for the simple task oflearning concepts corre(cid:173)\nsponding to axis-aligned rectangles in a multidimensional feature space. \nExisting learning models, when applied to this task, cannot explain how \nsubjects generalize from only a few examples of the concept. I propose \na principled Bayesian model based on the assumption that the examples \nare a random sample from the concept to be learned. The model gives \nprecise fits to human behavior on this simple task and provides qualitati ve \ninsights into more complex, realistic cases of concept learning. \n\n1 Introduction \n\nThe ability to learn concepts from examples is one of the core capacities of human cognition. \nFrom a computational point of view, human concept learning is remarkable for the fact that \nvery successful generalizations are often produced after experience with only a small number \nof positive examples of a concept (Feldman, 1997). While negative examples are no doubt \nuseful to human learners in refining the boundaries of concepts, they are not necessary \nin order to make reasonable generalizations of word meanings, perceptual categories, and \nother natural concepts. In contrast, most machine learning algorithms require examples of \nboth positive and negative instances of a concept in order to generalize at all, and many \nexamples of both kinds in order to generalize successfully (Mitchell, 1997). \n\nThis paper attempts to close the gap between human and machine concept learning by \ndeveloping a rigorous theory for concept learning from limited positive evidence and \ntesting it against real behavioral data. \nI focus on a simple abstract task of interest to \nboth cognitive science and machine learning: learning axis-parallel rectangles in ?Rm . We \nassume that each object x in our world can be described by its values (XI, ... , xm) on m \nreal-valued observable dimensions, and that each concept C to be learned corresponds to a \nconjunction of independent intervals (mini (C) ~ Xi ~ maXi (C\u00bb along each dimension \n\n\f60 \n\n(a) \n\n-\n\nr-------------. \n\nI \n\n+ \n\n+ \n\nI \nI \nI \nI \nI \n\n(b) \n\n(e) \n\n1. B. Tenenbaum \n\n...... ~ . \n\n\" \n\n- >\" ~ \n\n\\ - ...... - . ~ ~ \" \n\n\u2022 ! \n\n:C + \nI \nI \nt.. ... ____ ... __ ...... ___ , \n\n+ \n\nFigure 1: (a) A rectangle concept C. (b-c) The size principle in Bayesian concept learning: \nof the man y hypotheses consistent wi th the observed posi ti ve examples, the smallest rapidly \nbecome more likely (indicated by darker lines) as more examples are observed. \n\ni. For example, the objects might be people, the dimensions might be \"cholesterol level\" \nand \"insulin level\", and the concept might be \"healthy levels\". Suppose that \"healthy \nlevels\" applies to any individual whose cholesterol and insulin levels are each greater than \nsome minimum healthy level and less than some maximum healthy level. Then the concept \n\"healthy levels\" corresponds to a rectangle in the two-dimensional cholesterol/insulin space. \n\nThe problem of generalization in this setting is to infer, given a set of positive (+) and \nnegative (-) examples of a concept C, which other points belong inside the rectangle \ncorresponding to C (Fig. 1 a.). This paper considers the question most relevant for cognitive \nmodeling: how to generalize from just a few positive examples? \n\nIn machine learning, the problem of learning rectangles is a common textbook example \nused to illustrate models of concept learning (Mitchell, 1997). It is also the focus of state(cid:173)\nof-the-art theoretical work and applications (Dietterich et aI., 1997). The rectangle learning \ntask is not well known in cognitive psychology, but many studies have investigated human \nlearning in similar tasks using simple concepts defined over two perceptually separable \ndimensions such as size and color (Shepard, 1987). Such impoverished tasks are worth \nour attention because they isolate the essential inductive challenge of concept learning in a \nform that is analytically tractable and amenable to empirical study in human subjects. \n\nThis paper consists of two main contributions. I first present a new theoretical analysis \nof the rectangle learning problem based on Bayesian inference and contrast this model's \npredictions with standard learning frameworks (Section 2). I then describe an experiment \nwith human subjects on the rectangle task and show that, of the models considered, the \nBayesian approach provides by far the best description of how people actually generalize \non this task when given only limited positive evidence (Section 3). These results suggest \nan explanation for some aspects of the ubiquotous human ability to learn concepts from just \na few positive examples. \n\n2 Theoretical analysis \n\nComputational approaches to concept learning. Depending on how they model a con(cid:173)\ncept, different approaches to concept learning differ in their ability to generalize meaning(cid:173)\nfully from only limited positive evidence. Discriminative approaches embody no explicit \nmodel of a concept, but only a procedure for discriminating category members from mem(cid:173)\nbers of mutually exclusive contrast categories. Most backprop-style neural networks and \nexemplar-based techniques (e.g. K -nearest neighbor classification) fall into this group, \nalong with hybrid models like ALCOVE (Kruschke, 1992). These approaches are ruled out \nby definition; they cannot learn to discriminate positive and negative instances ifthey have \nseen only positive examples. Distributional approaches model a concept as a probability \ndistribution over some feature space and classify new instances x as members of C if their \n\n\fBayesian Modeling of Human Concept Learning \n\n61 \n\nestimated probability p(xIG) exceeds a threshold (J. This group includes \"novelty detec(cid:173)\ntion\" techniques based on Bayesian nets (Jaakkola et al., 1996) and, loosely, autoencoder \nnetworks (Japkowicz et al., 1995). While p(xIG) can be estimated from only positive ex(cid:173)\namples, novelty detection also requires negative examples for principled generalization, in \norder to set an appropriate threshold (J which may vary over many orders of magnitude for \ndifferent concepts. For learning from positive evidence only, our best hope are algorithms \nthat treat a new concept G as an unknown subset of the universe of objects and decide how \nto generalize G by finding \"good\" subsets in a hypothesis space H of possible concepts. \n\nThe Bayesian framework. For this task, the natural hypothesis space H corresponds to all \nrectangles in the plane. The central challenge in generalizing using the subset approach is \nthat any small set of examples will typically be consistent with many hypotheses (Fig. Ib). \nThis problem is not unique to learning rectangles, but is a universal dilemna when trying to \ngeneralize concepts from only limited positive data. The Bayesian solution is to embed the \nhypothesis space in a probabilistic model of our observations, which allows us to weight \ndifferent consistent hypotheses as more or less likely to be the true concept based on the \nparticular examples observed. Specifically, we assume that the examples are generated by \nrandom sampling from the true concept. This leads to the size principle: smaller hypotheses \nbecome more likely than larger hypotheses (Fig. Ib - darker rectangles are more likely), \nand they become exponentially more likely as the number of consistent examples increases \n(Fig. lc). The size principle is the key to understanding how we can learn concepts from \nonly a few positive examples. \nFormal treatment. We observe n positive examples X = {xCI), ... , x Cn )} of concept G \nand want to compute the generalization/unction p(y E GIX), i.e. the probability that some \nnew object y belongs to G given the observations X. Let each rectangle hypothesis h be \ndenoted by a quadruple (11,/2,81,82), where Ii E [-00,00] is the location of h's lower-left \ncomer and 8i E [0,00] is the size of h along dimension i. \nOur probabilistic model consists of a prior density p( h) and a likelihood function p( X I h) \nfor each hypothesis h E H. The likelihood is determined by our assumption of randomly \nsampled positive examples. In the simplest case, each example in X is assumed to be \nindependently sampled from a uniform density over the concept C. For n examples we \nthen have: \n\np(Xlh) \n\no otherwise, \n\n(1) \n\nwhere Ihl denotes the size of h. For rectangle (11,/2,81,82), Ihl is simply 8182. Note that \nbecause each hypothesis must distribute one unit mass oflikelihood over its volume for each \nexample cJx h p(xlh)dh = 1), the probability density for smaller consistent hypotheses is \ngreater than for larger hypotheses, and exponentially greater as a function of n. Figs. Ib,c \nillustrate this size principle for scoring hypotheses (darker rectang!es are more likely). \nThe appropriate choice of p( h) depends on our background knowledge. If we have no a \npriori reason to prefer any rectangle hypothesis over any other, we can choose the scale(cid:173)\nand location-invariant uninformative prior, p( h) = P(ll, 12, 81 ,82) = 1/(81,82), In any \nrealistic application, however, we will have some prior information. For example, we may \nknow the expected size O'i of rectangle concepts along dimension i in our domain, and then \nuse the associated maximum entropy prior P(ll, 12, 81,82) = exp{ -( 81/0'1 + 82/ 0'2)}. \nThe generalization function p(y E GIX) is computed by integrating the predictions of all \nhypotheses, weighted by their posterior probabilities p( h IX): \n\np(y E GIX) = r p(y E Glh) p(hIX) dh, \n\n(2) \n\nfrom Bayes' \n\nwhere \nthat \nfhEH p(hIX)dh = 1), and p(y E Clh) = 1 if y E hand 0 otherwise. Under the \n\n(normalized such \n\nlhEH \ntheorem p(hIX) \n\nex: p(Xlh)p(h) \n\n\f62 \n\nuninformative prior, this becomes: \n\nJ. B. Tenenbaum \n\n(3) \n\nHere ri is the maximum distance between the examples in X along dimension i, and \ndi equals 0 if y falls inside the range of values spanned by X along dimension i, and \notherwise equals the distance from y to the nearest example in X along dimension i. \nUnder the expected-size prior, p(y E GIX) has no closed form solution valid for all n. \nHowever, except for very small values of n (e.g. < 3) and ri (e.g. < 0'i/1O), the following \napproximation holds to within 10% (and usually much less) error: \n\n(4) \n\nFig. 2 (left column) illustrates the Bayesian learner's contours of equal probability of \ngeneralization (at p = 0.1 intervals), for different values of nand ri. The bold curve \ncorresponds to p(y E GIX) = 0.5, a natural boundary for generalizing the concept. \nIntegrating over all hypotheses weighted by their size-based probabilities yields a broad \ngradient of generalization for small n (row 1) that rapidly sharpens up to the smallest \nconsistent hypothesis as n increases (rows 2-3), and that extends further along the dimension \nwith a broader range ri of observations. This figure reflects an expected-size prior with \n0'1 = 0'2 = axiLwidthl2; using an uninformative prior produces a qualitatively similar plot. \nRelated work: MIN and Weak Bayes. Two existing subset approaches to concept learning \ncan be seen as variants of this Bayesian framework. The classic MIN algorithm generalizes \nno further than the smallest hypothesis in H that includes all the positive examples (Bruner \net al., 1956; Feldman, 1997). MIN is a PAC learning algorithm for the rectangles task, and \nalso corresponds to the maximum likelihood estimate in the Bayesian framework (Mitchell, \n1997). However, while it converges to the true concept as n becomes large (Fig. 2, row 3), \nit appears extremely conservative in generalizing from very limited data (Fig. 2, row 1). \n\nAn earlier approach to Bayesian concept learning, developed independently in cognitive \npsychology (Shepard, 1987) and machine learning (Haussler et al., 1994; Mitchell, 1997), \nwas an important inspiration for the framework of this paper. I call the earlier approach \nweak Bayes, because it embodies a different generative model that leads to a much weaker \nlikelihood function than Eq. 1. While Eq. 1 came from assuming examples sampled \nrandomly from the true concept, weak Bayes assumes the examples are generated by an \narbitrary process independent of the true concept. As a result, the size principle for scoring \nhypotheses does not apply; all hypotheses consistent with the examples receive a likelihood \nof 1, instead of the factor of 1/lhln in Eq. 1. The extent of generalization is then determined \nsolely by the prior; for example, under the expected-size prior, \n\n(5) \n\nWeak Bayes, unlike MIN, generalizes reasonably from just a few examples (Fig. 2, row 1). \nHowever, because Eq. 5 is independent of n or ri, weak Bayes does not converge to the \ntrue concept as the number of examples increases (Fig. 2, rows 2-3), nor does it generalize \nfurther along axes of greater variability. While weak Bayes is a natural model when the \nexamples really are generated independently of the concept (e.g. when the learner himself \nor a random process chooses objects to be labeled \"positive\" or \"negative\" by a teacher), it \nis clearly limited as a model oflearning from deliberately provided positive examples. \n\nIn sum, previous subset approaches each appear to capture a different aspect of how humans \ngeneralize concepts from positive examples. The broad similarity gradients that emerge \n\n\fBayesian Modeling of Human Concept Learning \n\n63 \n\nfrom weak Bayes seem most applicable when only a few broadly spaced examples have \nbeen observed (Fig. 2, row 1), while the sharp boundaries of the MIN rule appear more \nreasonable as the number of examples increases or their range narrows (Fig. 2, rows 2-3). \nIn contrast, the Bayesian framework guided by the size principle automatically interpolates \nbetween these two regimes of similarity-based and rule-based generalization, offering the \nbest hope for a complete model of human concept learning. \n\n3 Experimental data from human subjects \n\nThis section presents empirical evidence that our Bayesian model - but neither MIN nor \nweak Bayes - can explain human behavior on the simple rectangle learning task. Subjects \nwere given the task of guessing 2-dimensional rectangular concepts from positive examples \nonly, under the cover story of learning about the range of healthy levels of insulin and \ncholesterol, as described in Section 1. On each trial of the experiment, several dots \nappeared on a blank computer screen. Subjects were told that these dots were randomly \nchosen examples from some arbitrary rectangle of \"healthy levels,\" and their job was to \nguess that rectangle as nearly as possible by clicking on-screen with the mouse. The dots \nwere in fact randomly generated on each trial, subject to the constraints ofthree independent \nvariables that were systematically varied across trials in a (6 x 6 x 6) factorial design. The \nthree independent variables were the horizontal range spanned by the dots (.25, .5, 1, 2, 4, \n8 units in a 24-unit-wide window), vertical range spanned by the dots (same), and number \nof dots (2,3,4,6, 10,50). Subjects thus completed 216 trials in random order. To ensure \nthat subjects understood the task, they first completed 24 practice trials in which they were \nshown, after entering their guess, the \"true\" rectangle that the dots were drawn from. I \n\nThe data from 6 subjects is shown in Fig. 3a, averaged across subjects and across the two \ndirections (horizontal and vertical). The extent d of subjects' rectangles beyond r, the \nrange spanned by the observed examples, is plotted as a function of rand n, the number \nof examples. Two patterns of generalization are apparent. First, d increases monotonically \nwith r and decreases with n. Second, the rate of increase of d as a function of r is much \nslower for larger values of n. \n\nFig. 3b shows that neither MIN nor weak Bayes can explain these patterns. MIN always \npredicts zero generalization beyond the examples - a horizontal line at d = 0 - for all values \nof rand n. The predictions of weak Bayes are also independent of rand n: d = 0\" log 2, \nassuming subjects give the tightest rectangle enclosing all points y with p(y E G\\X) > 0.5. \nUnder the same assumption, Figs. 3c,d show our Bayesian model's predicted bounds on \ngeneralization using uninformative and expected-size priors, respectively. Both versions of \nthe model capture the qualitative dependence of d on rand n, confirming the importance of \nthe size principle in guiding generalization independent of the choice of prior. However, the \nuninformative prior misses the nonlinear dependence on r for small n, because it assumes \nan ideal scale invariance that clearly does not hold in this experiment (due to the fixed size \nof the computer window in which the rectangles appeared). In contrast, the expected-size \nprior naturally embodies prior knowledge about typical scale in its one free parameter 0\". A \nreasonable value of 0\" = 5 units (out of the 24-unit-wide window) yields an excellent fit to \nsubjects' average generalization behavior on this task. \n\n4 Conclusions \n\nIn developing a model of concept learning that is at once computationally principled and \nable to fit human behavior precisely, I hope to have shed some light on how people are able \n\nI Because dots were drawn randomly, the \"true\" rectangles that subjects saw during practice were \nquite variable and were rarely the \"correct\" response according to any theory considered here. Thus \nit is unlikely that this short practice was responsible for any consistent trends in subjects' behavior. \n\n\f64 \n\n1. B. Tenenbaum \n\nto infer the correct extent of a concept from only a few positive examples. The Bayesian \nmodel has two key components: (1) a generalization function that results from integrating \nthe predictions of all hypotheses weighted by their posterior probability; (2) the assumption \nthat examples are sampled from the concept to be learned, and not independently of the \nconcept as previous weak Bayes models have assumed. Integrating predictions over the \nwhole hypothesis space explains why either broad gradients of generalization (Fig. 2, row \n1) or sharp, rule-based generalization (Fig. 2, row 3) may emerge, depending on how \npeaked the posterior is. Assuming examples drawn randomly from the concept explains \nwhy learners do not weight all consistent hypotheses equally, but instead weight more \nspecific hypotheses higher than more general ones by a factor that increases exponentially \nwith the number of examples observed (the size principle). \n\nThis work is being extended in a number of directions. Negative instances, when encoun(cid:173)\ntered, are easily accomodated by assigning zero likelihood to any hypotheses containing \nthem. The Bayesian formulation applies not only to learning rectangles, but to learning \nconcepts in any measurable hypothesis space - wherever the size principle for scoring \nhypotheses may be applied. In Tenenbaum (1999), I show that the same principles enable \nlearning number concepts and words for kinds of objects from only a few positive exam(cid:173)\nples. 2 I also show how the size principle supports much more powerful inferences than \nthis short paper could demonstrate: automatically detecting incorrectly labeled examples, \nselecting relevant features, and determining the complexity of the hypothesis space. Such \ninferences are likely to be necessary for learning in the complex natural settings we are \nultimately interested in. \n\nAcknowledgments \n\nThanks to M. Bernstein, W. Freeman, S. Ghaznavi, W. Richards, R Shepard, and Y. Weiss for helpful \ndiscussions. The author was a Howard Hughes Medical Institute Predoctoral Fellow. \n\nReferences \n\nBruner, J. A., Goodnow,J. S., & Austin, G. J. (1956). A study of thinking. New York: Wiley. \n\nDietterich, T, Lathrop, R, & Lozano-Perez, T (1997). Solving the multiple-instance problem with \naxis-parallel rectangles. ArtificiaL Intelligence 89(1-2), 31-71. \n\nFeldman, J. (1997). The structure of perceptual categories. J. Math. Psych. 41, 145-170. \n\nHaussler, D., Keams, M., & Schapire, R (1994). Bounds on the sample complexity of Bayesian \nlearning using infonnation theory and the VC-dimension. Machine Learning 14, 83-113. \n\nJaakkola, T., Saul, L., & Jordan, M. (1996) Fast learning by bounding likelihoods in sigmoid type \nbelief networks. Advances in NeuraL Information Processing Systems 8. \n\nJapkowicz, N., Myers, C., & Gluck, M. (1995). A novelty detection approach to classification. \nProceedings of the 14th InternationaL Joint Conference on AritificaL InteLLigence. \n\nKruschke, J. (1992). ALCOVE: An exemplar-based connectionist model of category learning. Psych. \nRev. 99,22-44. \n\nMitchell, T (1997). Machine Learning. McGraw-Hill. \n\nMuggleton, S. (preprint). Learning from positive data. Submitted to Machine Learning. \n\nShepard, R (1987). Towards a universal law of generalization for psychological science. Science \n237,1317-1323. \n\nThnenbaum, J. B. (1999). A Bayesian Frameworkfor Concept Learning. Ph. D. Thesis, MIT \nDepartment of Brain and Cognitive Sciences. \n\n2In the framework of inductive logic programming, Muggleton (preprint) has independently \nproposed that similar principles may allow linguistic grammars to be learned from positive data only. \n\n\fBayesian Modeling of Human Concept Learning \n\n65 \n\nBayes \n\nMIN \n\nweak Bayes \n\nn=6 \n\nn= 12 \n\nFigure 2: Performance of three concept learning algorithms on the rectangle task. \n\n(a) Average data from 6 subjects \n\n(b) MIN and weak Bayes models \n\n52.5 \n\ni 2 \ne \n~ 1.5 \n& \n'0 1 \nC \n~ 0.5 \n\u2022\u2022 0 \n~ ~--~~--~----~------\n8 \n\n6 \n\no \n\n2 \n\n4 \n\nr: Range spanned by n examples \n\n2.5 \n\n2 \n\n1.5 weak Bayes (0 :: 2) \n\nweak Bayes (0:: 1) \n\nMIN \n\n0.5 \n\n0 \n0 \n\n2 \n\n4 \n\n6 \n\n8 \n\n\"In \n\n\"In \n\n\"In \n\n(c) Bayesian model (uninformative prior) \n2.5 \n\n(d) Bayesian model (expected-size prior) \n2.5 \n\n2 \n\n1.5 \n\no \n\n2 \n\n1.5 \n\nn::2 \nn::3 \nn=4 \nn=6 \nn\", 10 \n\nn= 50 \n\n2 \n\n4 \n\n6 \n\n8 \n\n2 \n\n4 \n\n6 \n\n8 \n\nFigure 3: Data from human subjects and model predictions for the rectangle task. \n\n\f\fPART II \n\nNEUROSCIENCE \n\n\f\f", "award": [], "sourceid": 1542, "authors": [{"given_name": "Joshua", "family_name": "Tenenbaum", "institution": null}]}