Learning Grounded Causal Models
Noah D. Goodman (ndg@mit.edu), Vikash K. Mansinghka (vkm@mit.edu), Joshua B. Tenenbaum (jbt@mit.edu)
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract
We address the problem of learning grounded causal models: systems of concepts that are connected by causal relations
and explicitly grounded in perception. We present a Bayesianframework for learning these models—both a causal Bayesian
network structure over variables and the consequential regionof each variable in perceptual space—from dynamic perceptual evidence. Using a novel experimental paradigm we showthat humans are able to learn grounded causal models, and thatthe Bayesian model accounts well for human performance.

Introduction
Imagine a toddler newly acquainted with the family cat, whosoon learns that petting causes purring. Prima facie this isa simple inference from observations: the constant conjunction of petting with purring leads to knowledge of the causal
relation between them. A number of approaches have madesubstantial progress in explaining how causal structure canbe learned in this way (Cheng, 1997; Waldmann and Martignon, 1998; Gopnik et al., 2004; Sloman, 2005; Griffiths
and Tenenbaum, 2005). However, all of these theories share
a common limitation: observable causal variables—the statesof the world that causal relations relate—must be supplied
from the start. Thus, our toddler finds an obstacle to applying any cognitive tools proposed by these theories: what is“petting”? That is, what unifies all of the perceptually different observed instances of petting, and separates them from
non-petting—and what makes petting a relevant causal variable? To an infant many aspects of the world might appear
as a “blooming buzzing confusion,” or at least a largely undifferentiated perceptual space. Yet that perceptual space issoon carved into the separate concepts that become the building blocks of causal understanding—the observable variables.
Where do observable variables come from? This question has
been raised in recent work on causal learning and cognitive
development (e.g. Gopnik et al., 2004), but not addressed ineither formal models or behavioral experiments. We begin to
do both in the following pages.
It could be that causal variables are not learned at all, butrather we have an innate endowment of variables and must
do the best we can with causal relations among them. However, considering the abstract and conceptual nature of many
variables, it seems likely that many are learned. If they arelearned, perhaps the simplest hypothesis for variable formation is bottom-up perceptual clustering: similar perceptions
are grouped together by early perceptual processes and only
later become available for causal learning. Indeed, it is wellknown that the perceptual system can perform sophisticatedclustering (e.g. gestalt laws of grouping), so this bottom-up
hypothesis may be a good description of learning in manycases. For instance, all of the perceptions of “explosions” arequite similar (or anyhow dissimilar to other percepts), so it
should be easy to cluster them together into a new variable,
which could later be used in causal learning (“gas leaks cause
explosions”).
However, people often make distinctions which are perceptually quite subtle but causally important. For example,
to our toddler petting and pounding the kitty may be perceptually similar—similar enough that they would be clustered
together by bottom-up grouping (e.g. simply “touching the
kitty”). In actuality, petting and pounding have very different
effects (purring versus hissing), and confounding them could
lead to problematic results from our toddler. Such examples
are the rule rather than the exception: laughing is ever-sosimilar to crying, but follows from different causes; pressing
the “volume” button on some T.V. remotes is very similar topressing the “channel” button, but has different effects. These
examples suggest, contrary to the bottom-up hypothesis, thatcausal structure is used to learn observable variables... but
how could the causal structure have been learned before thevariables? Instead, we must assume in these cases that variables are learned along with causal structure, so that variables
and structure may interact and constrain each other.
In this view, three layers of knowledge must be learned simultaneously (illustrated in Fig. 1): First, the number of variables and the possible states of each (e.g. petting and purring,which each have two states—happening or not). Second, thecausal structure amongst the variables (e.g. petting causes
purring). Third, the observation function for each variable,
which provides perceptual grounding into the world of thesenses: each is a map from perceptions to states of the variable. (Sounds with certain timbre and frequency, for example,
might be instances of purring.) Together, the number of variables, their states and observation functions, and the causal
structure, constitute a grounded causal model.
It is difficult to imagine how complete grounded causal
models could spring into being fully formed, Athena-like.
Indeed, the great virtue of the bottom-up hypothesis is its
comprehensible stages: perceptual clustering processes firstform variables, then causal learning mechanisms take overto discover causal structure1 . We will argue, however, thatinference of complete models can be described, at the computational level, in a Bayesian framework. Bayesian induction has been used to describe the learning of causal structure(Griffiths and Tenenbaum, 2005), and it naturally adapts todescribe the joint learning of structure and grounding. In thenext section we build a model of learning by combining asimple dynamic Bayesian network model of causal structure
1 We mean in particular bottom-up processes that separate variable formation from causal knowledge, for instance as an informationally encapsulated module.

with a “consequential region” model (Shepard, 1987) of observation functions. Because the number of variables is free
to vary, the model can learn as many, or as few, variables as
are useful in explaining the perceptual evidence.
After developing this model, we present a simple experimental paradigm to explore grounded causal learning whenneither the variables nor the causal structure are known inadvance. In this paradigm we describe three conditions, analogous to the petting/pounding example, which are indistinguishable to a bottom-up learner (see Fig. 1). We show thatpeople are able to successfully learn in this situation, andclearly distinguish the three conditions. Further, we find thatmodel predictions correlate well with human responses.

Modeling
In this section we introduce a Bayesian model for learning
grounded causal models, together with a specific example situation that forms the basis for our experimental tests. The
hypotheses of this model consist of three parts: the numberof variables, observation functions for the variables, and acausal structure over the variables. The observation functions
provide the relationship between the variables and perception, while the causal structure provides the relationship between the variables themselves. The machinery of Bayesian
probability gives a principled method to combine the separate pieces and to draw joint inferences about those pieces,balancing complexity against the ability to explain perceptual data. In the remainder of this section we describe eachpiece of the model, then assemble the pieces into a likelihood
function describing the probability of a sequence of percepts;
finally, Bayes’ rule is used to invert this dependency, giving
the posterior probability of each grounded causal model.
We start with a space of possible perceptual configurations2 in any given instant, P . An observable variableconsists of a set of states, S, and an observation function,f : P → S, mapping each point in the perceptual space to astate of the variable. We will focus on the case of binary
variables (i.e. S={0, 1}), in which the observation function isdetermined by a consequential region (Shepard, 1987)—thepre-image of the “active” state (i.e. f −1 (1)).
Take a simple example world of small dots appearing on arectangular screen. For a screen with a single dot the perceptual space is given by screen coordinates: [0, 1]2 . If the number of dots can vary, fromnone up to a maximum of M, theS
2mperceptual space is: P = M
m=0 [0, 1] . In this world a useful
set of observation functions is given by “buttons”: rectangular regions that are active when there is a dot inside them. If
r ⊆ [0, 1]2 is such a rectangle, and w=(w1 , ..., wk )∈P is a percept, then theW“button” observation function corresponding tor is fr (w) = km=1 δwm ∈r (that is, 1 if and only if at least oneof the k dots is in the rectangle).
Returning to the general situation, say that we have a set of
N variables, vi , with corresponding observation functions fi .
2 We leave open the question of what level of perception thisspace represents, for instance egocentric or allocentric.

A causal structure C relates the sequence of states si,t = fi (wt )observed from a sequence of percepts wt ∈P . We assume
that causal relations hold only between cause variables at
one instant and effect variables at the next (that is, we assume that causality is dynamic—causes precede effects—andMarkov—states only depend directly on states in the previous
instant). Thus the causal structure C is a graph on the variables such that the direct parents of variable vi , indicated byparC (vi ), are its causes. For simplicity we assume a simple
nearly-deterministic-or parametrization: a state is active at agiven instant, si,t =1, if any parent v j ∈par(vi ) is active at theprevious instant, s j,t−1 =1, or with some small probability ε(and similarly for off). In addition, any state may be madeactive by an intervention: si,t =1 if Inti,t =1. Putting together
this causal structure3 : 1

if Inti,t =1, or∃v j ∈parC (vi ) s.t. s j,t−1 =1,P(si,t =1|st−1 ,C, Int) = εotherwise.
1+ε(1)
Next, the observation function fi of each variable determines a consequential region f −1 (1), where the variable isactive, and its complement f −1 (0), where the variable is inactive. There is a region of perceptual spacecompatible withT
any set of states st of the variables: Rst = i fi−1 (si,t ). We assume that percepts of a given state are drawn uniformly fromthis region. Thus the likelihood of a particular percept wt ,conditioned on the observation functions f and states st , is:1+εP(wt |st , f ) =δ f (wt )=st.
|Rst |(2)This likelihood leads to a size principle (Tenenbaum and Griffiths, 2001): a percept is assigned higher probability when it
falls in a smaller region R f (w) . Critically, this provides inductive pressure to select variables that minimize the average
size of the regions R f (wt ) —cutting regions of perceptual space
which are frequently visited into finer pieces than those which
are rare.
For the example world described above, interventions
should be thought of as taps (or mouse-clicks) on the screen,
which activate a “button” if they fall within it. We make one
minor adjustment to Eq. 2 for this situation: if a variable ismade active by an intervention then there must be a dot at
the site of the intervention, when this happens P(wt |st , f )=1.
This has an interesting consequence: a variable which is madeactive only by interventions (e.g. a variable with no causal
parents) will exhibit no size principle effect. We return to thisprediction in the next section.
The full likelihood comes from Eqs. 1 and 2, by marginalizing over states, and recalling that each state is independent
3 We will simplify subscripts of quantities like state s so that sti,t
indicates the vector of states at time t, and s indicates all states at all
times.

Causal
Structure:

Observation functions

Perceptual
Grounding:Consequential regions

Perceptual Space

(a)

(b)

of others at the same instant:NP(wt |st−1 ,C, f , Int) = ∑ P(wt |st , f ) ∏ P(si,t |st−1 ,C, Int).
st

i=1

(3)
However, since Eq. 2 is zero for all but the “observed” state
sob
i,t = f i (wt ), this simplifies to:obP(wt |st−1,C, f , Int) =

1 N
ob
P(sob
i,t |st−1 ,C, Int).
|Rsob | ∏
i=1

(4)

t

For a sequence of perceptions w with observed states stob (weassume that all initial states sobi,-1 =0):Tob
P(w| f ,C, Int) = ∏ P(wt |st−1,C, f , Int).

(5)

t=0

By Bayes rule, the posterior probability of a grounded causal
model f ,C is:P( f ,C|w, Int) ∝ P( f ,C)P(w| f ,C, Int).

(6)

For simplicity, and in order to focus on intrinsic aspects ofthe model, we use a uniform prior on number of variables,
causal structure, and observation functions: P( f ,C)∝1. (The
uniform prior on number of variables is not proper, but is regularized by the likelihood.) To account for memory limitations on the sequence of percepts, and possible discountingof earlier information, we include a power-law decay in thelikelihood term (T is the last time):T−γP( f ,C|w, Int) ∝ ∏ P(wt |st−1 ,C, f , Int)(T −t)

(7)

t=0

Experiment
To investigate human abilities to learn grounded causal models when neither the variables nor the causal structure are
known in advance, we adopted the simple perceptual spacedescribed above. Participants interacted, by clicking freely,with a blank rectangular window on a computer screen, which
they were told was an “alien panel”. In response to these interventions a sequence of one or more dots would sometimes
bloom and then disappear at various locations on the window.
The laws underlying the dots’ appearance, unknown to the

Figure 1: Grounded causal
models: number of variables,
causal structure, and observation functions. Panels (a) and(b) show models that cover
the same regions of perceptual space, so could not be
distinguished by bottom-up
clustering.

participants, were similar to the example used earlier: a number of invisible rectangular “buttons” served as variables (theconsequential region of each identical to its physical extent
on the screen). These buttons were related by a deterministic
dynamic causal structure, in which a button was active if any
parent was active or if an intervention click was made within
it. An active button created a dot within its (invisible) boundary. (As above, a click inside a button always resulted in a dot
at the site of the click.)
We wished to create an experimental situation, similar to the petting/pounding example discussed in the introduction, in which two variables had very similar perceptual properties—so would be indistinguishable by bottom-up
grouping—but different causal properties. Because the consequential regions of the buttons are rectangular, a set of dots
occurring within a single large button occupies the same perceptual space as dots appearing at random in two small buttons that subdivide the large one (see Fig. 1). Thus the threestructures illustrated at the left of Fig. 2, which occupy thesame regions of perceptual space, should be indistinguishable
to a purely bottom-up learner. To put it another way, if one
simply clusters the dots appearing within these regions there
is no reason to split adjacent rectangles into separate clusters.
However, without splitting the adjacent rectangles it is impossible to correctly learn causal structures a or b. (Of course aclustering algorithm that takes advantage of causal information could distinguish these conditions—the Bayesian model
described above can be seen as such an algorithm.)Thus we had three experimental conditions: one for eachstructure/condition shown in the left of Fig. 2. With this design we wished to test two main hypotheses: First, people
can learn grounded causal models—learning structure andgrounding at the same time, and distinguishing conditions impossible for a bottom-up learner. Second, the structure andgeometry of the models people infer are consistent with predictions of the Bayesian model.

MethodParticipants Participants were 17 members of the MIT
community. Two participants failed to understand the instructions, and were excluded from further analyses.

structure a:

(i)

(ii)

structure b:
a

b

c

(iii)

1

1

1

0.5

0.5

0.5

0

(iv)

a

b

c

0

(v)

a

b

c

0

(vi)

structure c:

a

b

c

1

1

1

0.5

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0

Figure 2: Experimental conditions (structures a, b, and c), and examples of individual participants’ observations and responses(i–vi). Participants’ interventions are marked by crosses, and resulting dots are marked by dots. Participants’ responses are insolid red lines, and the actual structures (not seen by participants) are in dashed blue. Inset in each example is the evidencespecific posterior probability of the model. (i–iii) are correct responses, (iv–vi) are incorrect, as predicted by the model.
Materials and Procedure All interactions, responses, andinstructions were presented on a computer screen4 . Participants first read a brief cover story: “Scientists have discovered alien artifacts that look like blank panels, but respond ininteresting ways when clicked upon.” Participants were thengiven a brief familiarization with the “alien panel” screen,
and with the drawing tool. The alien panel was blank, except
for light grey grid lines, and responded to clicks, as describedabove, with sequences of blue dots. The drawing tool looked
similar, but also had a tool pallet that enabled participants todraw rectangles and arrows. During familiarization participants were told that “scientists have determined that thesepanels have ‘active regions’,” and were shown two active regions made visible “by a special process.”
There were three “brand new alien panels,” one for eachof the conditions described above (see Fig. 2, left). Participants interacted with each alien panel for five blocks of approximately 30 clicks each. Each block was followed bya chance to “describe what’s going on” by using the drawing tool. Thus each participant made fifteen responses (three
conditions, five blocks), each of which was a diagram freely
drawn using rectangles and arrows (see Fig. 2, right, for examples of participants’ responses). The order of the three
conditions was counter-balanced across participants, and the
4 An online demonstration of the experiment can be found at:
http://www.mit.edu/˜ndg/LGCMdemo.html

physical orientation of the panels was randomized.
Model approximation and fitting Continuous coordinates
were approximated on a fine grid (equal to the pixel widthof the screen). Posterior probabilities were evaluated by enumerating over a large subset of hypotheses, including the actual structures and reasonable alternatives that varied in number of variables and causal structure. (Hence the ratios of posterior probability values reported are exact, while the absolutemagnitudes are approximate.) The two free parameters of themodel, capturing causal transmission noise and “forgetting”
rate, were adjusted by hand to ε=0.3, γ=0.5.

Results
We investigate the causal structure and geometry of participants’ responses. Before turning to statistical analyses, let us
consider a few example responses. In Fig. 2(i-iii) we see theresponses of three participants, one in each condition, whocorrectly identified the underlying causal structure. The rectangular “active regions” are often drawn by participants offset or slightly mis-sized relative to their actual locations. Thismay be due in part to the difficulty of remembering over thecourse of the experiment the precise location of interventions
and dots. Participants also often separated adjacent variables
in conditions A and B, as in Fig. 2(i)—this is suggestiveof overregularization due to categorical perception, though
could also be due to pragmatic issues with the drawing tool.

Many “errors” made by participants are, in fact, rational responses to available evidence, and predicted by the Bayesianmodel. For instance, errors in early blocks are often failures
to discover all of the active regions—that is, they are the result of insufficiently broad interventions, see Fig. 2(iv). Occasionally, though all regions have been explored, the evidence generated still favors the wrong hypothesis (according
to the model), leading people to a “rational error”. For instance, in Fig. 2(v) we see a participant in condition A who
has drawn structure c—which is also favored by the model(inset). Of course not all errors made by participants are easily explained—Fig. 2(vi).
To enable group analyses, each response was coded as
structure a, b, c, or “other” by two coders who were blind
to condition and block. (Responses were coded in random
order to preclude influence of other responses of the same
participant). Coders were instructed to code a response as a,b, or c only if it had the correct number of variables, in approximately the correct locations, and correct causal arrows.
There was 96% agreement between the two coders; differences were resolved by discussion. The portion of participants responding with each structure, for each condition andblock, are shown in Fig. 3 (black bars). By the final block themodal human response in each condition was correct—that
is, the modal response was the actual structure for that condition. Thus participants were able to learn each groundedcausal model, given sufficient interaction time. Moreover,participants strongly distinguished the three conditions, providing significantly different response patterns between theconditions (χ2 (6)=133.5, p<0.001, aggregating responses
across blocks). Since these conditions are indistinguishable
to a bottom-up learner, this result makes it quite unlikely thathuman learning of grounded causal models is purely bottomup.
We next want to explore whether the structure and geometry of participants’ responses are consistent with the Bayesianmodel. Since participants generated their own interventions,each observed unique evidence; it is thus necessary to apply the model separately to the evidence available to eachparticipant at the time of each response. The inset plots ofFig. 2 show these evidence-specific model posterior probabilities. The mean posterior probabilities for each condition andblock, averaged across evidence-specific results for individual participants, are shown in Fig. 3 (white bars). Comparing
the human response rates (black bars) to the model predictions reveals quite good qualitative agreement. Notice, forinstance, that the model correctly predicts that condition A
is harder than conditions B or C, and that in condition A asignificant minority of responses are structure c. The quantitative fit is also good: correlation r=0.95, and mean deviation 0.04.
To see whether errors on individual responses werepredicted by the model we investigated evidence-specificmodel posterior probabilities for correct responses (those thatmatched the actual structure) and incorrect responses. The

posterior probability of the actual structure is significantly
lower for incorrect trials than correct trials (one-tailed MannWhitney U test, U=2877, N=225, p<0.0001). This suggeststhat the evidence-specific posterior is a good predictor of human errors—and thus that many of these errors are actually
rational responses to the available evidence.
Turning to the geometry of the “active regions”—that is, toinferences about properties of the observation functions—weconsider the proximity of causes and effects and their relative
sizes. In structure a, the ‘effect’ variables were spatially adjacent while the ‘cause’ variables were separated; vice versa
in structure b. Participants correctly inferred this aspect of
the geometry: in structure a responses the ‘cause’ variables
were drawn by participants significantly closer together thanthe ‘effect’ variables (t(56)=5.58, p<0.0001), and in structure b responses the ‘effects’ were significantly closer thanthe ‘causes’ (t(58)=9.38, p<0.0001).
Recall that, because ‘cause’ variables can only be activatedby interventions and an intervention created a dot exactly at
the location of the click, only the ‘effect’ variables should
be subject to a size principle under the model. For the ‘effect’ variables the size principle implies that hypotheses withsmaller active regions will have higher probability. Hence,
the model predicts the areas for ‘cause’ variables to be larger,
on average, than those for ‘effect’ variables. This prediction
was validated for both structure a responses (t(114)=3.57,
p<0.001) and structure b responses (t(118)=2.88, p<0.005).

Discussion and Conclusion
Where do observable variables come from? We have suggested that observable causal variables are learned along withthe causal structure that relates them, with causal structure influencing the discovery of variables and variables grounding
the causal structure in perception. We developed a Bayesianmodel to describe the inference of grounded causal models
by combining simple Bayesian networks for causal structurewith consequential region observation functions. A simple
example of this model led to a novel experimental paradigm
for the learning of grounded causal models—an aspect of
causal learning that has not been previously investigated. Ina behavioral test using this paradigm humans were able tocorrectly learn grounded causal models, and to distinguishconditions that should be indistinguishable to a bottom-up,
clusters-then-causes, learner. Further, the model was shown
to be a good predictor of the structures and geometry foundby human participants, predicting errors on both the groupand individual level.
However, our experimental results are only a preliminary
test of the model. Further investigation will be needed toevaluate whether learned variables are truly coherent abstract
concepts, and to explore the subtle trade-offs between causal
structure and observation functions. The model can also be
refined in several important ways. First, our simple modelof causal structure should be extended to include inhibitory,
interactive, and noisy causal relationships. Second, the set

Block 1

Block 2Block 3Block 4Block 5

Condition A

1

0.5

Condition B

0
1

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Condition C

0
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0

a

b

c

a

b

c

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b

c

of observation functions we used, while well suited to theexperimental situation, is too simple for many real-world settings. It may be possible to adapt existing models of perceptual discrimination, such as fragment-based recognition(Ullman et al., 2001), to provide flexible observation functions. Third, intervention was treated casually; abstract interventions should be grounded in physical actions, and thisgrounding must also be learned. Beyond simple refinementsof this model a more precise and detailed model will eventually be needed: we’ve only begun to scratch the surface ofwhat could be modeled in this domain—leaving open, for instance, the process and time course of learning, and the generation of exploratory interventions.
We have recently proposed a model for learning causal
types (Kemp et al., 2007), an important kind of abstract causal
knowledge. (This model was applied it to data from the experiment of Lien and Cheng (2000) in which unknown causal
structure interacted with unknown causal types—intriguingly
similar to the way that grounding and structure interacted inour experiment.) It is worth noting that the model of Kemp
et al. (2007) and the one proposed in this paper can be combined to provide a computational model of causal learningthat spans from perceptual grounding to abstract knowledge.
Such a combination is especially interesting because it suggests a novel kind of abstract causal knowledge: frameworklevel prior beliefs about observation functions for new variables. For instance, adults use prior knowledge to quickly
infer that a new doorbell-shaped-patch next to a door, evenone never before seen, is likely to be a causal variable. Priorknowledge about observation functions is likely at work inmany adult inferences of new causal variables.
The ideas of this paper fit in a broader theme in cognitive
science and philosophy. In order for concepts to be useful
they must have a semantic grounding: a manner of connecting
an internal representation of the concept to the external world.
An influential theory of semantic grounding holds that themeaning of a concept is its observation conditions (ie. statesof the world in which it is true). The semantic groundingof concepts has been addressed a number of times in psy-

a

b

c

a

b

c

Figure 3: Black bars: theportion of human responses
consistent with each structure (a, b, and c; shown atleft), for each condition
and block. White bars:
mean model posterior
probabilities. Correlation
between the two is r=0.95.

chology, notably in work on feature formation (Schyns et al.,1998). On the other hand, a great deal of the meaning of aconcept could come from its conceptual role: the internal relationships it has with other concepts. The theory-theory, especially in its causal models incarnation (Gopnik et al., 2004),
has influentially developed this component of concept meaning. By considering grounded causal models we have begun
to explore a notion of concepts in which meaning comes fromboth sources: the observation conditions for concepts, and thecausal relations between them. By describing how groundedcausal models can be learned from perception we have, perhaps, started to unravel how human minds come to have suchrich, meaningful, concepts.

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