Learning Structured Generative Concepts
Andreas Stuhlmüller, Joshua B. Tenenbaum, Noah D. Goodman
Brain and Cognitive Sciences, MIT{ast, jbt, ndg}@mit.edu
Abstract
Many real world concepts, such as “car”, “house”, and “tree”,
are more than simply a collection of features. These objects
are richly structured, defined in terms of systems of relations,subparts, and recursive embeddings. We describe an approachto concept representation and learning that attempts to capturesuch structured objects. This approach builds on recent probabilistic approaches, viewing concepts as generative processes,
and on recent rule-based approaches, constructing concepts inductively from a language of thought. Concepts are modeled
as probabilistic programs that describe generative processes;
these programs are described in a compositional language. In
an exploratory concept learning experiment, we investigate human learning from sets of tree-like objects generated by processes that vary in their abstract structure, from simple prototypes to complex recursions. We compare human categorization judgements to predictions of the true generative process aswell as a variety of exemplar-based heuristics.

Introduction
Concept learning has traditionally been studied in the context of relatively unstructured objects that can be describedas collections of features. Learning and categorization can beunderstood formally as problems of statistical inference, anda number of successful accounts of concept learning can beviewed in terms of probabilistic models defined over different
ways to represent structure in feature sets, such as prototypes,
exemplars, or logical rules (Anderson, 1990; Shi, Feldman,& Griffiths, 2008; Goodman, Tenenbaum, Feldman, & Griffiths, 2008). Yet for many real world object concepts, such as“car”, “house”, “tree, or “human body”, instances are morethan simply a collection of features. These objects are richly
structured, defined in terms of features connected in systemsof relations, parts and subparts at multiple scales of abstraction, and even recursive embedding (Markman, 1999). A tree
has branches coming out of a trunk, with roots in the ground;branches give rise to smaller branches, and there are leaves
at the end of the branches. A human body has a head on topof a torso; arms and legs come out of the torso, with arms
ending in hands, made of fingers. A house is composed ofwalls, roofs, doors, and other parts arranged in characteristic
functional and spatial relations that are harder to verbalize but
still easy to recognize and reason about. Besides objects, examples of structured concepts can be found in language (e.g.
the mutually recursive system of phrase types in a grammar),in the representation of events (e.g. a soccer match with its
fixed subparts), and processes (e.g. the recipe for making apancake with steps at different levels of abstraction).
Such concepts have not been the focus of research in theprobabilistic modeling tradition. Here we describe an approach to representing structured concepts—more typical of
the complexity of real world categories—using probabilistic

generative processes. We test whether statistical inferencewith these generative processes can account for how people
categorize novel instances of structured concepts and compare with more heuristic, exemplar-based approaches.
Because a structured concept like “house” has no single,
simple perceptual prototype that is similar to all examples,learning such a concept might seem very difficult. However,each example of a structured concept itself has internal structure which makes it potentially very informative. Consider
figure 1, where from only a few observations of a concept it
is easy to see the underlying structural regularity that can beextended to new items. The regularities underlying structured
concepts can often be expressed with instructions for generating the examples: “Draw a sequence of brown dots, choosea branch color, and for each brown dot draw two dots of thiscolor branching from it.”Figure 1: Three examples of a structured concept described by asimple generative process.

We build on the work of Goodman, Tenenbaum, et al.
(2008), who introduced an approach to concept learning as
Bayesian inference over a grammatically structured hypothesis space—a “language of thought.” Single concepts expressed in this language were simple propositional rules forclassifying objects, but this approach naturally extends toricher representations, providing a concept learning theory
for any representation language. Here we consider a language
for generative processes based on probabilistic programs: instructions for constructing objects, which may include probabilistic choices, thus describing distributions on objects—in
our case distributions on colored trees. Because this language
describes generative processes as programs, it captures regularities as abstract as subparts and recursion.
The theory of concept representation that we describe here
shares many aspects with previous approaches to concepts.
Like prototype and mixture models (Anderson, 1990; Griffiths, Canini, & Sanborn, 2007), probabilistic programs describe distributions on observations. However, prototypes andmixtures generate observations as noisy copies of ideal prototypes for the concept and thus cannot capture more abstract
structures such as recursion. Like rule-based models of concept learning, our approach supports compositionality: complex concepts are composed out of simple ones—but rather

Formal Framework
In the following, we first explain the formal language we use
to describe generative processes, then the different methodsof categorization (or generalization) we compare to subjects’
judgements.

´

We analyze concepts as generative models, i.e. as formal descriptions of processes that generate observations. We do so
within a simple domain where we can fully know and manipulate the actual generating processes behind complex objects.
We use tree-structured graphs with colored nodes as observations in our experiments—these are a simple proxy for many
real-world concepts, where the dependencies among parts arehierarchical or tree-like. Human bodies, buildings, and events
all consist of parts that themselves contain parts, with eachpart standing in interesting relation to the others.
We represent these trees as nested lists: each list denotes
a tree, with the first element in the list specifying the colorof the root node and the remaining elements describing thechildren of this node, each child itself being a list (tree). For
example, the second tree shown in figure 1 can be represented
as (• (•) (• (•) (•)) (•)).
We formalize the processes that generate these observations using a subset of Church, a Lisp-like stochastic programming language1 (Goodman, Mansinghka, Roy,
Bonawitz, & Tenenbaum, 2008). Programs in Church describe processes that produce values; running a program corresponds to generating a value from such a process. Because
Church contains primitive functions that randomly choose
from a distribution on values (e.g. the function flip that randomly chooses true or false), Church programs describe
stochastic processes. The meaning of a Church program is
1 Church uses prefix notation, i.e. function application is writtenwith the operator first, the operands following. For example, (nodex y) means that the function node is called with the arguments x and
y.

´

´

Concept Representationa distribution on return values—which may be complex values such as nested lists—and any given execution results ina sample from this distribution. In what follows we describe
Church programs which sample colored trees.
We group generative models into classes by the abstract
constructions they use. Table 1 illustrates each of these types
using a single concept program and observations drawn fromthis program. The simplest tree-generating processes in our
language use only the stochastic function node, which takes
as its first argument a color symbol and as its remaining arguments subtrees. With high probability, node returns a treethat has the given color symbol at its root and the given subtrees as its children, but with some probability ε, it switches
to a noise process that can return any tree, that is, node introduces a random noise process into the tree construction.
Under the noise process, the number of children for a node issampled from a geometric distribution with parameter ε andthe node color is sampled uniformly.
Programs like (node • (node •) (node •)) denotestochastic prototypes. They are most likely to generate the tree that corresponds to the given colors, in thiscase (• (•) (•)), but they can return any tree with a certain
probability. The more a tree deviates from the prototype, theless likely this process is to generate it. For example, thesimple program described above could switch at the third
node to the noise process and produce (• (•) (• (•)))
instead of the prototype. By introducing the noise process,node turns a deterministic prototype into a stochastic process.
All of the more abstract ways of formalizing generativemodels in our tree domain compose these basic processes.
Nested prototypes formalize the intuition that a concept or apart of a concept can be “either this or that”. Running the
program (if (flip .5) (node •) (node •)) will flip a faircoin and return a sample from (node •) with probability .5,
otherwise a sample from (node •).
One of the central reasons for analyzing concepts asrepresented in a language of thought is that they compose analogously to the components of natural and artificial
languages—parts similarly allow composition through reusein our domain. A part concept is defined first and can thenbe used in arbitrarily many places within other concepts. For
example, the program (define (part) (node • (node •)))
names a simple part consisting of only two nodes. Thispart can now be reused in other concepts. For example,
the most likely return value for (node • (part) (part))is (• (• (•)) (• (•))). When parts are defined, they areavailable to the noise process. This leads to some invarianceto the position of parts and captures the idea that a generatingprocess may give rise to observations that contain a part in adifferent place, although with lower probability compared toan observation with the part in the correct place.
Parameterized parts can capture both deterministic structure and random choices and reuse them in multiple places.
When a part like (define (part x) (node • x x)) is used,
for example in the program (part (node •)), it evaluates´than deterministic rules, our concepts denote distributions.
Finally, the probabilistic program approach can be seen as a
generalization of previous approaches to generative representations of concepts (Kemp, Bernstein, & Tenenbaum, 2005;
Rehder & Kim, 2006; Feldman, 1997).
We investigate human learning for classes of generatingprocesses that vary in their abstract structure, from simple
prototypes to complex multiply recursive programs. We compare predictions for categorization judgments based on thetrue generative model to the predictions of exemplar models,
which exploit the relational structure of the examples to varying degrees but cannot detect more abstract structure. We
find two regimes: for concepts with simple prototype-likestructures, human judgements are well described by a relational exemplar model, but humans can also easily learn more
abstract regularities—such as sub-concepts and recursion—
which are better captured by a model using more expressive
generative descriptions based on probabilistic programs.

´

the body of the part—here (node • x x)—with x assigned toits argument, here (node •). Evaluating the program (part(node •)) is therefore most likely to result in the observation (• (•) (•)).
Allowing parts to call themselves introduces recursion,a means to capture a large amount of repetitive observed
structure in a single short definition. For example, the part
(define (p) (if (flip) (node •) (node • (p)))) can generate arbitrarily deep lists of single blue nodes, with shorter
ones being more likely.
The power of these program constructs is that they may be
used compositionally to build more complex concepts, suchas those shown in table 1.

Categorization
In order to model generalization and categorization behavior of human subjects, we need not only a way to represent
concepts, but also a way to compute the probability of any
given observation belonging to a known concept. We analyze
our experimental results using four models that differ in how
much they make use of representational structure.
On the unstructured end of the scale, we use a model thatcomputes generalization judgements solely by comparing thefraction of nodes that have a given color. On the other end of
the scale, a generative Bayesian model uses the likelihood under the true generative process to judge category membership.
In between, an exemplar model makes use of tree structure in
the observations, but not of the more abstract generative process that led to the observations.
Generative Model In modeling concept learning as
Bayesian program induction, we follow the approach taken
by Goodman, Tenenbaum, et al. (2008). Since we formalize
concepts as probabilistic programs, the likelihood P(O|C) ofan observation O under a given concept C corresponds to theprobability of the program making its random choices suchthat it returns the observation as its value (see Goodman,
Mansinghka, et al. (2008)). The posterior probability of aconcept C given observations O is proportional to this likelihood multiplied by the prior:
P(C|O) ∝ P(O|C)P(C)(1)

In the last section, we described a language for programs
which generate trees; a prior P(C) could be derived from thislanguage, as in Goodman, Tenenbaum, et al. (2008). An ideal
learner would then infer the posterior distribution P(C|O)
over concepts C given the observation O and make predictions about whether a new observation t belongs to the category of the observed objects using each concept C ∈ C inproportion to its posterior probability:
P(t|O) ∝ ∑ P(t|C)P(C|O)(2)

C

In order to make computational modeling tractable, we
make the simplifying assumptions that (1) subjects’ reasoning
is dominated by the maximum a posteriori (MAP) estimate ofthis distribution, i.e. by the single concept that has the highest
posterior probability and that (2) the true generating concept
Ctrue is a good approximation to the MAP estimate. Thus, foreach of the concept types we investigate, we model subjects’
behavior using the program from which the training data was
sampled. The likelihood of a new observation t belonging tothis concept is simply P(t|Ctrue ) which we compute using an
adaptive importance sampling algorithm.
We do not claim that subjects necessarily identify the true
generating concept from a few examples; this approximation
is made for computational tractability. The full Bayesianmodel, which maintains uncertainty over generating concepts, can make different predictions in certain cases, but it
is not clear whether this represents a bias for or against theapproximation—to the extent that people remain uncertain of
the concept after a few examples, the Bayesian model would
capture human inferences better than our approximation.
Tree Exemplar Model This and the next two models are
versions of the exemplar-based generalized context model(GCM) (Nosofsky, 1986). For observations O1 , . . . , On fromcategory C and a new observation t for which we would
like to estimate the likelihood under category C, we use
P(t ∈ C|O1 , . . . , On ∈ C) ∝ n1 ∑ni=1 e−d(Oi ,t) where d is a distance measure that is sensitive to the tree structure of the observations. Starting from the root node, this measure matches
the trees as much as possible, incrementing by 1 for each nodethat differs in color between the two trees and for each nodethat must be generated because it exists in one tree but not in
the other tree. This approach is similar to the structure mapping approach used by Tomlinson and Love (2006).
Frequency-based Exemplar Models As in the tree exemplar model, we use a distance measure d to estimate the likelihood of an observation belonging to a category for which
we have only positive examples. In this version of the model,d(t1 ,t2 ) is the RMSE between the transition count vectors of
t1 and t2 . For each pair of node colors, the transition countvector contains the number of times this pair occurs adjacent(as parent-child) in the given tree. We call this model Transition GCM. We also investigate a simplified version that uses
the distance between the color count vectors. The length of
this vector corresponds to the number of possible node colors,with each entry in the vector denoting how often this nodecolor appears in the tree of interest. We call this Set GCM.

ExperimentThis experiment is an exploratory investigation into generalization from observations of structured objects. Since ourmain goal in this study is to investigate the representationof concepts and their use for categorization and generalization rather than the memory aspects of learning, we use aparadigm that minimizes memory demands. By doing so,
we hope to focus on how people represent the commonalities
between observed instances of a concept and how they use
this knowledge to generalize to new instances. We chose aPrototype

Nested Prototype(node •
(node •
(node •
(node •)
(node •))))(node •
(node •
(node •
(if (flip .5)(node •
(node •)
(node •
(node •
(node •))
(node •)))
(node •
(node •)
(node •
(node •))
(node •))))))

Parts(define (part)(node •
(node •
(node •))))(node •
(part)(node •
(node •
(part))(part)))

ParameterizedParts(define (part x)(node •
x
(node •
x
(node •
x
(node • x x)
x)
x)
x))
(part
(if (flip .5)(node •)
(node •)))

Single Recursion(define (part)(node •
(if (flip .5)(node •
(part)(node •))
(node •))))(node •
(node •
(node •)
(node •))
(part))

Multiple
Recursion(define (part)(node •
(if (flip .3)(part)(node •))
(if (flip .3)(part)(node •))))(node •
(node •
(node •
(part)))(part))

Table 1: This table illustrates the concept types that can be represented within our language for generative models. For each type, an example
of a concept (a stochastic program) is shown together with observations drawn from this program. The stochastic function node generates
a mixture of the subtrees that are passed to it as its arguments and a noise process that, with low probability, can generate any tree. The
abstraction methods stochastic branching, (parameterized) parts and recursion compose these stochastic prototypes into more structured
generative processes.

domain that both contains observations with simple structure
and allows for interesting generative processes—the domainof colored trees generated by probabilistic programs.

Methods
Participants 250 members of Amazon’s crowdsourcing
service Mechanical Turk took part in the online experiment.
Subjects were compensated for participation.
Stimuli Subjects were told that they are looking at newly
discovered kinds of plants that grow in extreme environments.
Each subject saw 18 pages, with each page consisting of 15
training examples, a control question, and a test example together with a classification question. Both training and test
examples were images of simple trees with colored nodes
drawn from tree-generating programs (see e.g. table 3). For
each of the concept types shown in table 1, there were threetree-generating programs, and for each program there were 7
test examples. These test examples were chosen to cover awide range of both intuitive and model judgements of category membership. Both training example order and stimuli
colors were randomized.

Procedure In order to ensure that subjects process the training stimuli, a control question on each page asked how manyof the training trees consist of more than 7 dots. 55 subjects
answered less than 13 out of the 18 control questions correctly within an error margin of 2. We did not include thesesubjects in the analysis.
The categorization question asked: “How likely is it thatthe following plant is the same kind of plant as the plants
above?” Subjects chose on a seven-step scale ranging from“certainly the same kind” to “certainly not the same kind”.
For each subject, the responses were normalized to [0, 1].

Results
Table 2 summarizes the correlation results for all models.
Figure 2 shows for each concept type human results andmodel results for both the exemplar and generative model.
For each concept type, three different concepts were part ofthe experiment, and for each concept, seven different test observations were shown. A single point in the scatterplot contains information on the mean subject response for a single
test tree and on the model prediction for this tree.
Neither of the two exemplar models based on simple statis-

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Prototype

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Figure 2: Comparison between human and model responses across concept types for tree exemplar and generative model. For each of thesix concept types, three examples were shown; the color of the dots indicates to which example any given datapoint belongs. Empty circles
denote isolated part cases that were excluded from the correlation analysis.

Prototype
Nested Prototype
Parts*
Parameterized
Parts
Single Recursion
Multiple
Recursion

Set
GCM
0.589
0.544
0.320
0.298

Transition
GCM
0.751
0.851
0.617
0.591

Tree
GCM
0.803
0.937
0.705
0.778

Generative
Model
0.748
0.904
0.835
0.911

0.284
0.505

0.499
0.561

0.637
0.451

0.773
0.770

Table 2: Human-model correlations for the experiment. Each row
shows how well the different models predicted subjects’ performance for a particular concept type. *Correlations excluding isolated part cases (see text).

tics was the best predictor for any of the concept types, withthe transition-based exemplar model performing strictly better than the set-based model. An effect that is not accounted
for by the less structural exemplar models is illustrated by
the nested prototype example in table 3: Subjects generalize
significantly more to examples with branches they have seen
before than to examples that have a mixture of two known
branches. Likewise, subjects seem to generalize significantly
more to trees with known branches than to trees that have newbranches with similar surface statistics. Both results are expected under the two models that make use of tree structure.
If we group prototype and nested prototype as “less structured” and subconcepts with and without arguments, single
recursions, and multiple recursions as “more structured”, then
the tree exemplar model best predicts human responses forthe less structured stimuli whereas the true generative model
best predicts performance for the more structured stimuli.
Our generative model makes the simplifying assumptionthat the learner infers a single generating concept from the examples whereas one interpretation of the tree exemplar modelis that it uses each of the training examples as a hypothesison what the true concept looks like. A fully Bayesian learner,
which maintains a distribution over generative processes, may

predict human behavior in ways similar to the tree exemplar
model for less structured examples and similar to the true generating process model for the more structured examples.
Having seen how different models predict human judgements for different concept types, we will now look at individual response patterns in order to determine ways in which
both of the two structural models can be improved.
The part example in table 3 shows how changes to the location of a part can have significantly different effects depending on whether the overall concept is preserved (resulting in high generalization) or the part is moved into a completely different environment (resulting in low generalization). By analogy, a Picasso face, with eyes in odd places,
is still more of a face than an eye alone. Parts seen out ofcontext constitute a problem for all models (except for thesimplest set-based one): subjects judged these isolated partsas unlikely to come from the concept that included them as
subparts whereas the models did give a high score to these examples. Since including these outliers dramatically changed
the scores and made the interpretation of the model comparison difficult, we excluded these data points from the analysis in table 2. Without correction, the model-human correlations for the part concepts are: 0.403 for the set-based exemplar model, 0.505 for the transition exemplar model, 0.512
for the tree-based exemplar model, and 0.543 for the generative model (note that rank-order among the models does notchange as a result of excluding these data points).
For the parameterized part example in table 3, changing theargument uniformly, i.e. in all places where it occurs, leads
to consistently higher scores than changing the argument differently in different places; however, this difference is not
significant. This difference is expected if subjects inferred
the true generative model, since changes to the argument require only one use of the noise process, whereas nonuniformchanges require many different nodes to be generated by thenoise process. Future research needs to determine whether
this effect is real, perhaps by manipulating the diversity ofConceptType

Nested Prototypes

Parts

Parameterized
Parts

Single Recursion

Multiple
Recursion

...

...

...

...

...

Training

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**

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***

Table 3: This table illustrates a small selection of our experimental results. For five different concept types, training observations from asingle concept of this type are shown together with subjects’ generalizations for particularly interesting test examples. The error bars are
standard errors of the mean.

parameter arguments in the observations.
For the single recursion example in table 3, changing thecolor of a few nodes within the recursion results in a significantly lower generalization. At the same time, a very similar manipulation does not result in a significant change in
the generalization rating for the multiple recursion example.
Intuitively, we sometimes see a change as destroying a very
obvious pattern structure whereas at other times, the change
in structure is not assumed to be relevant. Future researchneeds to characterize when subjects infer that such a pattern
exists, and when they instead assume coincidence.
The comparison between the frequency based exemplar
models and the two models that rely on tree structure in theobservations makes clear that subjects do make use of thefact that the observations are structured in their generalization judgements. Furthermore, comparing the tree exemplar
model to the true generative model that makes use of more abstract structure hints at the possibility that subjects are relying
on recursive structure in the observations. The individual response patterns in the results of our exploratory experimenthighlight ways in which both the exemplar-based model andthe generative model can be improved to more closely reflect
human generalization patterns.

Conclusion
Most studies of concept learning have focused on relatively
unstructured objects based on simple features. We have suggested viewing concepts as probabilistic programs that describe stochastic generative processes for more structured objects. In this view concepts denote distributions over objects,
and these distributions are built compositionally. We exploredthis idea within a domain of tree-like objects, and carried out
a study of human generalization using a broad variety of con-cepts in this domain. Our results suggest that humans areable to extract abstract regularities, such as recursive structure, from examples, but also that there are many subtle effects to be discovered and accounted for in such domains.
Acknowledgements We thank Frank Jäkel and Brenden
Lake for useful comments. This work was funded in partby grants from the ONR (N00014-09-0124) and the AFOSR(FA9550-07-1-0075).

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