Learning Causal Schemata
Charles Kemp, Noah D. Goodman & Joshua B. Tenenbaum{ckemp, ndg, jbt}@mit.edu
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Abstract
Causal inferences about sparsely observed objects are often
supported by causal schemata, or systems of abstract causal
knowledge. We present a hierarchical Bayesian frameworkthat learns simple causal schemata given only raw data as input. Given a set of objects and observations of causal events involving some of these objects, our framework simultaneously
discovers the causal type of each object, the causal powersof these types, the characteristic features of these types, andthe characteristic interactions between these types. Previous
behavioral studies confirm that humans are able to discovercausal schemata, and we show that our framework accounts
for data collected by Lien and Cheng and Shanks and Darby.
Keywords: causal induction; intuitive theories; hierarchical
Bayesian models; categorization

Introduction
People often make accurate causal inferences based on very
sparse data. Imagine, for instance, that you are travelling in
the tropics, and on your very first morning you take an antimalarial pill and wash it down with guava juice. Soon afterward you develop a headache and wonder what might havecaused it. Suppose that you have very little direct information
about the two potential causes—you have never before tasted
guava juice or taken anti-malarial pills. Even so, you willprobably correctly attribute your headache to the pill rather
than the juice.
Accurate inferences from sparse data often rely on the topdown influence of abstract knowledge. Even if you havenever come across anti-malarial pills or guava juice, youprobably know about the causal powers of pills in generaland juices in general—in particular, you know that pills tend
to cause headaches but that juices do not. Abstract causal beliefs of this sort are sometimes called causal schemata [6] orintuitive theories.
Two fundamental questions can be asked about causal
schemata: how do these schemata support top-down inferences given relatively sparse data, and how are theseschemata acquired? This paper develops a hierarchical
Bayesian framework that provides a unified approach to bothquestions. Griffiths [4] has previously shown that hierarchical Bayesian models help to explain how top-down inferences
can be guided by causal schemata. Here we focus on the acquisition question, and show that hierarchical Bayesian models help to explain how causal schemata can be acquired bybottom-up learning.
Top-down and bottom-up approaches to causal learning aresometimes seen as competitors. The top-down approach [6,14] emphasizes inferences that are based on knowledge aboutcausal powers, and the bottom-up approach emphasizes statistical inferences that are based on patterns of covariation.
As Cheng [2] and others have argued, these perspectives are
best regarded as complementary: top-down knowledge aboutcausal power plays a role in many inferences, and bottom-up
statistical learning can help to explain how this knowledge
is acquired. The apparent conflict between these perspectives may have developed in part because there is no wellestablished framework that accommodates them both. Kelley,
for example, argued for both top-down [6] and bottom-up approaches [7] to causal reasoning, but did not develop a single
theoretical framework that properly unified his two proposals.
We will argue that a hierarchical Bayesian approach provides
this missing theoretical framework, and will develop a modelthat shows how schemata support causal reasoning and how
schemata can be acquired by statistical learning.
Part of our task is to formalize the notion of a causal
schema. Suppose that we are interested in a set of objects—
for example, a set of pills. This paper works with schemata
that assign each object to a causal type, and specify the causal
powers and features of each type. Our pills, for instance, may
represent four causal types—pills of type A cause headaches,
pills of type B relieve headaches, and pills of types C and Dneither cause nor relieve headaches. A causal schema may
also specify how causal types interact. For instance, a C-pill
and a D-pill may cause a headache when taken together, eventhough neither pill causes a headache on its own.
The first section of this paper considers the well-studied
problem (Fig. 1a) of learning a causal model that captures
the relationship between a single object (e.g. a pill) and an
effect (e.g. a headache). Causal models for several objects
can be learned independently, but this approach ignores any
information that might be shared across objects: for instance,
two blood-pressure medications are likely to have similar side
effects, enabling us to predict that a new blood-pressure medication will cause headaches if several others already have.
The second section introduces causal schemata that groupthe objects into types, and specify the likely causal powersof the objects belonging to each type (Fig. 1b). We show
how these schemata can be acquired in settings where learners must learn a schema at the same time as they are learningcausal models for many different objects.
By tracking the characteristic features of causal types,
learners can often make strong predictions about a novel object before it is observed to participate in any causal interactions. For instance, predictions about a pill with a given color,size, shape and imprint can be based on the effects produced
by previous pills which shared these features. The third sec-

(a)

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Figure 1: (a) A generative framework for discovering thecausal powers of a single object. (b) A generative framework
for learning a schema that guides inferences about multiple
objects. The schema organizes the objects into causal types,
and specifies the causal powers of each type. (c) A generative
framework for learning a schema that includes information
about the characteristic features of each type. Concrete examples of each framework are shown in Figs. 2b, 2c, and 3.
tion extends the notion of a causal schema by including information about the characteristic features of each causal type(Fig. 1c). Although we begin with cases where at most one
object is present at any time, the final section considers cases
where multiple objects may be present. We extend the notionof a schema one more time by allowing interactions between
different types (for instance, pills of type C may interfere withpills of type D), and we show how these characteristic interactions can be learned.

Learning a single causal model
Suppose that we are interested in the relationship between asingle object o and an effect e, and that we have observed acollection of event data V . Each observation in V represents
a trial where the object o was either present or absent andthe effect e either was or was not observed. For instance, if
object o is a pill and effect e is a headache, each trial might
indicate whether or not a patient took a pill on a given day,and whether or not she subsequently experienced a headache.
To simplify our notation, o will refer both to the pill and to
the event of the patient swallowing the pill.
We assume that the outcome of each trial is generated froma causal model M that captures the causal relationship between o and e (Figs. 1a and 2b). Having observed the event
data V , our beliefs about the causal model can be summarized
by the posterior distribution
P (M |V ) ∝ P (V |M )P (M ).

(1)

We build on the approach of Griffiths and Tenenbaum [5] andparameterize the causal model M using four causal variables(Fig. 2a and 2b). Let a indicate whether there is an arrow
joining o and e, and let g indicate the polarity of this causal
relationship (g = 1 if o is a generative cause and g = 0 if o
is a preventive cause). Suppose that s is the strength of therelationship between o and e.1 To capture the possibility that1

To simplify the later development of our model, we assume thatg and s are defined even if a = 0 and there is no causal relationshipbetween o and e. When a = 0, g and s can be interpreted as thepolarity and strength that the causal relationship between o and e
might have had if this relationship actually existed.

e will be present even though o is absent, we assume that agenerative background cause of strength b is always present
We specify the distribution P (e|o) by assuming that generative and preventive causes combine according to a network ofnoisy-OR and noisy-AND-NOT gates.
Now that we have parameterized model M in terms of thetriple (a, g, s) and the background strength b, we can rewrite
Equation 1 asp(a, g, s, b|V ) ∝ P (V |a, g, s, b)P (a)P (g)p(s)p(b).

(2)

To complete our framework we must place prior distributions
on the four causal variables. We use uniform priors on the twobinary variables (a and g), and assume that the two continuous variables (s and b) represent the logistic transformationsof Gaussian variables drawn from conjugate priors.2 We set
the hyperparameters of these conjugate priors to encourage b
to be small and s to be large.

Learning multiple causal models
Suppose now that we are interested in a set of objects {oi }and a single effect e. We begin with the case where at mostone object is present at any time: for example, suppose thatour patient takes many different pills, but at most one per day.
Instead of learning a single causal model our goal is to learn
a set {Mi } of causal models, one for each pill (Figs. 1b and2c). There is now a triple (ai , gi , si ) describing the causal
model for each pill oi , and we collect these variables into
three vectors, a, g and s. Let Ψ be the tuple (a, g, s, b) which
includes all the parameters of the causal models. As for the
single object case, we assume that a generative backgroundcause of strength b is always present.
Instead of learning each causal model separately, we introduce the notion of a schema. A schema specifies a groupingof the objects into causal types, and indicates the causal powers of each of these types. The schema in Fig. 2c indicatesthat there are two causal types: objects of type t1 tend to prevent the effect, and objects of type t2 tend to cause the effect.
Formally, let zi indicate the type of oi , and let ā, ḡ, and s̄ beschema-level analogues of a, g, and s: ā(t) is the probability
that any given object of type t will be causally related to theeffect, and ḡ(t) and s̄(t) are the expected polarity and causal
strength for objects of type t. Even though ā and ḡ are vectors of probabilities, Fig. 2c simplifies by showing each ā(t)and ḡ(t) as a binary variable.
To generate a causal model for each object, we assume
that each arrow variable ai is generated by tossing a coinwith weight ā(zi ), that each polarity gi is generated by tossing a coin with weight ḡ(zi ), and that each strength si isdrawn from the logistic transform of a Gaussian distribution with mean s̄(zi ) and variance σ̄(zi ). Let Ψ̄ be the tuple2

For instance, logit(s) is drawn from a Gaussian with mean µ
and variance σ 2 . σ 2 is drawn from an inverse gamma distribution
with shape parameter a and scale parameter b, and µ is drawn froma Gaussian with mean m and variance vσ 2 . We set v = 10, a = 2
and b = 0.3 for all continuous variables. For strength variables, we
set m = 1, and for the background variable we set m = −1.

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Figure 2: (a) Causal graphical models which capture two possible relationships between an object o and an effect e. a indicates
whether there is a causal relationship between o and e, g indicates whether this relationship is generative or preventive, ands indicates the strength of this relationship. A generative background cause of strength b is always present. A third possiblemodel (a = 1, g = 0) is not shown. (b) Learning a causal model M from event data V (see Fig. 1a). The event data specify the
number of times the effect was (e+ ) and was not (e− ) observed when o was absent and when o was present. The model shown
has a = 1, g = 1, s = 0.9 and b = 0.2, and is an instance of the second model in (a). (c) Learning a schema and a set of causal
models (see Fig. 1b). z specifies a set of causal types, where objects belonging to the same type have similar causal powers,
and ā, ḡ, and s̄ specify the causal powers of each type. Note that the schema supports inferences about an object (o7 ) that isvery sparsely observed.
(ā, ḡ, s̄, σ̄). To complete the model, we specify prior distributions on z and Ψ̄. We use a prior P (z) that assigns someprobability mass to all possible partitions but favors partitions
that use a small number of types.3
Having defined a generative model, we can use it to learn
the type assignments z, the schema parameters Ψ̄ and the parameters Ψ of the causal models that are most probable given
the event data V we have observed:p(z, Ψ̄, Ψ|V ) ∝ P (V |Ψ)P (Ψ|Ψ̄, z)p(Ψ̄|z)P (z).

(3)

Fig. 2c shows how a schema and a set of causal models (top
two rows) can be simultaneously learned from the event data
V in the bottom row. All of the variables in the figure havebeen set to values with high posterior probability according toEquation 3: for instance, the partition z shown is the z withmaximum posterior probability. Note that learning a schema
supports confident inferences about object o7 , which is very
sparsely observed (see the underlined entries in Fig. 2c). On
its own, a single trial might not be very informative about thecausal powers of a novel object, but experience with previous
objects allows our model to predict that o7 will produce theeffect as regularly as the other members of type t2 .
To compute the predictions of our model we implementeda Markov chain Monte Carlo scheme that samples from theposterior distribution in Equation 3. Our implementation,
however, is not intended as a process model, and our primary contribution is the computational theory summarized by
Equation 3.
3
We use a Chinese Restaurant Process prior on P (z), and set
the concentration parameter to 1. The entries in ā and ḡ are independently drawn from a Beta(0.1, 0.1) distribution, and the meansand variances in s̄ and σ̄ are drawn from the conjugate prior alreadydescribed.

Learning causal types given feature data
Imagine that you are allergic to nuts, and that one day you
discover a small white sphere in your breakfast cereal—amacadamia nut, although you do not know it. To discover
the causal powers of this novel object you could collect somecausal data—you could eat it and wait to see what happens.
Probably, however, you will observe the features of the object(its color, shape and texture) and decide to avoid it since it
is similar to other allergy-producing foods you have encountered.
Our formal framework naturally handles the idea that instances of a given causal type tend to have similar features(Figs. 1c and 3). Suppose that we have a matrix F which
captures many features of the pills in our study, including
their sizes, shapes, colors, and imprints. We assume that objects belonging to the same type have similar features. For
instance, the schema in Fig. 3 specifies that objects of type t1
tend to have feature f7 but not f8 . Formally, let the schemaparameters Ψ̄ include a matrix F̄ , where f¯j (t) specifies the
expected value of feature fj within causal type t.4 Building
on previous models of categorization [1], we assume that thevalue of fj for object oi is generated by tossing a coin withbias f¯j (zi ). Our goal is now to use the features F along withthe event data V to learn a schema and a set of causal models:p(z, Ψ̄, Ψ|V, F ) ∝ P (V |Ψ)P (F |Ψ̄, z)p(Ψ|Ψ̄, z)p(Ψ̄|z)P (z).
There are many previous models for discovering categories
of objects with similar features [1], and feature-based categorization is sometimes pitted against causal categorization [3].
Our framework works with the idea that real-world categories
are often distinguished both by their characteristic features
4
The prior on F̄ assumes that all entries in this matrix are independent draws from a Beta(0.5, 0.5) distribution.

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Figure 3: (a) Learning a schema and a set of causal models given event and feature data (see Fig. 1c). Objects belonging to thesame type have similar causal powers and similar features, and f¯i specifies the expected value of feature fi within each type.
The event and feature data shown are for the horizontal condition of the Lien and Cheng experiment. (b) A summary of thefeature matrix shown in (a). Feature f7 is shared by all and only the first eight objects, and f9 is shared only by the first four
objects. (c) Event data for two conditions. 10 trials were shown for each of the first 12 objects. In the horizontal condition, eachobject with feature f7 produces the effect on 8 out of 10 trials. In the vertical condition, only objects with f1 regularly produce
the effect. (d) Predictions for the sorting task of Lien and Cheng [9]. The first two rows show the percentage of subjects whogrouped a novel object (o13 , o14 or o15 ) with the f1 -match (o1 ) rather than the f8 -match (o10 ). Only subjects in the verticalcondition tend to sort according to f1 . The model predictions represent the relative probability that each novel object belongs
to the same causal type as the f1 -match.
and their characteristic causal interactions. More often thannot, one kind of information will support the categories indicated by the other, but there will also be cases where thecausal data and the feature data conflict. In cases like this, ourmodel may discover the feature-based categories, the causal
categories, or some combination of both—the categories preferred will depend on the relative weights of the statisticalinformation present in the two kinds of data.

Behavioral data
Lien and Cheng [9] ran several experiments that explore how
perceptual features and causal observations can both informcausal judgments. Our framework can handle all of theirtasks, but we focus here on a simplified version of their firsttask. The effect of interest is whether a certain kind of plant
blooms, and the potential causes are 15 chemicals (objects o1
through o15 ). Fig. 3b shows that the features of these objects(f1 through f14 ) support two systems of categorization. The
first is based on color: each object has a cool color (f7 ) ora warm color (f8 ), and the warm-colored objects are either
yellow (f11 ), red (f12 ) or orange (f14 ). Similarly, each object
has an irregular shape (f1 ) or a regular shape (f2 ) and there
are three kinds of irregular shapes (f13 , f3 and f4 ).
We show our model 10 trials for each of the first 12 objects, and Fig. 3c summarizes the results of these trials. In

the horizontal condition, each object with a cool color (f7 )causes blooming on 8 out of 10 occasions, and the remaining
objects lead to blooming less often. In the vertical condition,
objects with irregular shapes (f1 ) are the only ones that tend
to cause blooming. In both conditions, the model is shownthat blooming occurs on 2 out of 10 trials when no chemicals
are applied.
We test our model by requiring it to reason about threeobjects (o13 , o14 and o15 ) for which no trials were observed(see the underlined entries in Fig. 3a). Object o13 has a novel
shape, o15 has a novel color, and o14 is a novel combination ofa known shape and known color. Each novel object was presented as part of a trio that also included o1 and o10 , and wecomputed whether the model preferred to group each novel
object with the shape match (o1 ) or the color match (o10 ).5
In the horizontal condition, the model prefers to sort each trio
according to color (f8 ), but in the vertical condition the model
sorts each trio according to shape (f1 ) (see Fig. 3d). Notethat the feature data and the causal data must be combined to
produce this result: a model that relied on the features alone
would predict no difference between the two conditions, and5
We implemented this sorting task by computing the posterior
distribution p(z|V, F ), and comparing the probability that the novel
object and its color match belong to the same causal type with theprobability that the novel object is grouped with the shape match.

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Figure 4: Learning about interactions between objects. The schema specifies the causal powers of each type and of each combination of types (the combination t1 +t2 ) is not shown. The collection of causal models includes a model for each combinationof objects. The event data are inspired by the experiment of Shanks and Darby [13]. The model groups the objects into two
types: objects belonging to type t1 cause the effect on their own but not when paired with each other, and objects belonging to
the type t2 cause the effect only when paired with each other.
a model that used only the causal data would be unable to
make useful predictions about the three novel objects. Since
we have modeled a simplified version of the Lien and Chengtask, the quantitative predictions of our model are not directlycomparable to their results, but Fig. 3d shows that our model
captures the main qualitative patterns in their data.6

Discovering interactions between causal types
So far we have considered problems where at most one object
oi can be present at a time. Suppose now that multiple objects
can be present on any trial. For instance, consider the problem of discovering which drugs produce a certain allergy—
two drugs which are innocuous on their own may produce theallergy when combined. Our goal is to discover a schema anda set of causal models that allow us to predict whether any
given combination of drugs is likely to produce an allergic
reaction. Formally, we would like to learn a causal model M
for each possible combination of objects.
We assume that each combination of objects corresponds
to a conjunctive cause that may be generative or preventive,and extend Ψ to include an arrow a, a polarity g and a strengths for each combination of objects. We extend the schema in asimilar fashion, and include schema parameters ā, ḡ, s̄ and σ̄
for each combination of causal types. The causal model parameters for sets of objects are generated, as before, from theschema parameters for the corresponding set of types. For
instance, Fig. 4 shows how the causal model for o13 +o14 isgenerated from the schema-level knowledge that pairs of objects drawn from type t2 tend, in combination, to generate theeffect with strength 0.9. As before, we assume that a generative background cause of strength b is always present.
There are several possible strategies for handling conjunctive causes and our approach makes several simplifying as6
Lien and Cheng report that a handful of subjects did not groupthe novel objects with either the shape match or the color match.
These subjects were dropped before computing the percentages inFig. 3d.

sumptions. For instance, we assume that the causal powerof a conjunction of objects is independent of the causal powers that correspond to any subset of these objects. To accurately capture human intuitions, it will be necessary to relax
our simplifying assumptions, and to combine our frameworkwith a sophisticated approach to conjunctive causality [11].
Here, however, we have aimed to provide the simplest possible example of how our framework can discover interactions
between causal types.

Behavioral data
Shanks and Darby [13] ran an experiment which suggeststhat humans can acquire abstract knowledge about interactions between causal types. These authors used a task where
the potential causes were foods, and the effect of interest wasan allergic reaction. The data observed by participants in theirsecond experiment are shown in Fig. 4.7 When supplied withthese data, our model discovers two causal types: foods oftype t1 (o1 through o8 ) produce the allergy on their own, but
foods of type t2 (o9 through o16 ) do not. The model also discovers that two foods of type t2 will produce the allergy wheneaten together, but two foods of type t1 will not (Fig. 4).
Shanks and Darby were primarily interested in predictions
about cases which had never been observed in training—thecases underlined in Fig. 4. Their participants can be divided
into two groups according to their scores when tested on the
training data. Learners in the high group (learners who scoredwell on the test) tended to make the same predictions as ourmodel: for instance, they tended to predict that o7 and o8 produce the allergy when eaten in isolation, that o15 and o16 do
not, that the combination of o13 and o14 produces the allergy,
and that the combination of o5 and o6 does not. Learners in
the low group tended to make the opposite predictions: for
instance, they tended to predict that o7 and o8 do not produce the allergy when eaten in isolation. Since our compu7
Different subjects saw different amounts of training data, but
we overlook this detail.

tational framework does not suffer from memory limitations
or lapses of attention, it is not surprising that it accounts only
for the predictions of learners who absorbed the information
provided during training.

Discussion
We described a hierarchical Bayesian framework (Fig. 1c) for
learning causal schemata. Our hierarchical framework supports several kinds of inferences. We focused on bottom-up
learning and showed that the model helps to explain how acausal schema and a set of specific causal models can be simultaneously learned given event data and feature data. If
the causal schema is known in advance, then the framework
serves as a computational theory of top-down causal learning,and explains how inferences about a set of causal models cansimultaneously draw on low-level event data and top-downknowledge.
Our work exploits the fact that probabilistic approaches
are modular and can be composed to build integrated models of inductive reasoning. The model in Fig. 1c can be created by combining three models: probabilistic causal models [12] specify how the event data are generated given a setof causal models, the infinite relational model [8] specifies
how the causal models are generated, and Anderson’s rational approach to categorization [1] specifies how the features
are generated. Since all three models work with probabilities it is straightforward to combine them to create a single
integrated framework for causal reasoning.
We showed that our framework helps to explain some aspects of the data collected by Lien and Cheng [9] and Shanks
and Darby [13], and it also accounts for several other resultsin the literature. Waldmann and Hagmayer [16] showed that aknown set of categories can influence future causal learning,and our approach predicts a similar result if we fix the causal
types z then use our framework to discover a set of causal
models given event data. Our framework can also model experiments carried out using the blicket detector [3] or causal
blocks world [15] paradigms. Many aspects of these experiments have been previously modeled, but our framework
captures phenomena that are not addressed by most existingmodels. For instance, our model suggests why two identical looking blocks might both be categorized as blickets eventhough a handful of observations suggest that they have different effects on a blicket detector [3].
Several extensions of our approach may be worth exploring. We restricted ourselves to problems where the distinction between a set of potential causes and a set of effects 8
is known in advance, but in some cases this distinction mightneed to be learned [10]. A second limitation is that we focused on cases where feature data and contingency data represent the only input to our model. Human learners are sometimes directly supplied with abstract causal knowledge—for
example, a science student might be told that “pineapple juice
8This paper has focused on problems where there is a single effect, but our approach also handles problems with multiple effects.

is an acid, and acids turn litmus paper red.” Statements likethese correspond to fragments of a causal schema, and future
experiments should explore how schemata are learned whenparts of these schemata are directly supplied.
More often than not, competing accounts of a given phenomenon both capture some element of the truth. Where
possible, cases like these should be handled by building unified accounts that subsume the two competing views. We
have developed a hierarchical Bayesian model that attempts
to unify top-down and bottom-up approaches to causal reasoning. Similar conflicts between top-down and bottom-up
approaches are found in other areas of cognitive science, andthe hierarchical Bayesian approach may be useful for resolving these conflicts wherever they occur.
Acknowledgments Supported by the William Asbjornsen
Albert memorial fellowship (CK), the James S. McDonnell
Foundation Causal Learning Collaborative Initiative (NDG,
JBT) and the Paul E. Newton chair (JBT).

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