Learning a Theory of Causality
Noah D. Goodman, Tomer D. Ullman, Joshua B. Tenenbaum{ndg, tomeru, jbt}@mit.edu
MIT, Dept. of Brain and Cognitive Sciences
Abstract
We consider causality as a domain-general intuitive theory
and ask whether this intuitive theory can be learned from cooccurrence of events. We begin by phrasing the causal Bayes
nets theory of causality, and a range of alternatives, in a logical language for relational theories. This allows us to explore
simultaneous inductive learning of an abstract theory of causality and a causal model for each of several causal systems. We
find that the correct theory of causality can be learned relatively quickly, often becoming available before specific causal
theories have been learned—an effect we term the “blessing of
abstraction”. We then explore the effect of providing a variety of auxiliary evidence, and find that a collection of simple“input analyzers” can help to bootstrap abstract knowledge.
Together these results suggest that the most efficient route tocausal knowledge may be to build in not an abstract notion of
causality, but a powerful inductive learning mechanism and avariety of perceptual supports. While these results are purely
computational, they have implications for cognitive development, which we explore in the conclusion.
Keywords: Causality, hierarchical Bayesian model, innateness.

Introduction
What allows us to extract stable causal relations from thestream of experience? Hume believed that it was the principle of association: constant conjunction of events follow froman underlying association; from this principle, and observed
events, one may infer a causal association. Recent researchin psychology (Pearl, 2000; Gopnik et al., 2004) and philosophy (Woodward, 2003) has established the interventionist
or causal Bayes nets (henceforth CBN) account of causality
as a description of the principles by which causal reasoning
proceeds. These principles include a directed, probabilistic
notion of causal dependence, and a privileged role for uncaused manipulation—the interventions, which include actions and experimental manipulations. The CBN frameworkleads to quicker, more reliable learning than weaker assumptions about the nature of causation, and has been successful
at predicting human learning (e.g. Gopnik et al., 2004). In
recent discussions of the psychological basis of causality it isoften assumed that the principles of CBN, or some variant,
are an innate resource. In this paper we will argue there isan alternative: that the principles guiding causal understanding in humans can be seen as an intuitive theory, learnablefrom evidence out of more primitive representations. Our argument, which will proceed via an ideal learner analysis, canbe seen as both an investigation into the psychological basisof causality, and a case-study of abstract learning and the roleof innate structure in Bayesian approaches to cognition.
We have previously proposed that intuitive theories—
systems of abstract concepts and laws relating them—can be
represented in a “language of thought” which includes as-

pects of probability and logic (Kemp, Goodman, & Tenenbaum, 2008; Tenenbaum, Griffiths, & Niyogi, 2007; Goodman, Tenenbaum, Griffiths, & Feldman, 2007). Because theassumptions of CBN are formalizable via probability andlogic, they are potentially expressible in such a language for
intuitive theories. This suggests the hypothesis that CBN
is not an innate resource, but is itself an intuitive theory of
causality, learned inductively from evidence and represented
in a more basic language of theories.
A theory of causality would have several properties unusual for an intuitive theory. First, it would be domaingeneral knowledge. Intuitive theories are typically thought ofas domain-specific knowledge systems, organizing our reasoning about domains such as physics or psychology, butthere is no a priori reason to rule out domain-general knowledge. Second, a theory of causality would have to be acquired
remarkably early in development. There are intriguing hints
that aspects of causal knowledge are present from early infancy, but little evidence that a full notion of cause is innate(Saxe & Carey, 2006). Yet if a theory of causality is to underly the acquisition of specific causal knowledge it must be
available within the first year of life. Could such an abstract
theory be learned from evidence so rapidly, even in principle?
To investigate this question we turn to hierarchical Bayesianmodeling.
The formalism of hierarchical Bayesian modeling makes
it possible to express the assumptions relating knowledge at
multiple levels of abstraction (Gelman, Carlin, Stern, & Rubin, 1995), and Bayesian inference over such a model describes an ideal learner of abstract knowledge (Tenenbaum,
Griffiths, & Kemp, 2006). Though real learning is undoubtedly resource-constrained, the dynamics of an ideal learner
can uncover unexpected properties of what it is possible tolearn from a given set of evidence. For instance, it has
been reported (e.g. Kemp, Perfors, & Tenenbaum, 2007) thatlearning at the abstract level of a hierarchical Bayesian model
is often surprisingly fast in relation to learning at the more
specific levels. We term this effect the blessing of abstraction1 : abstract learning in an HBM is often achieved before
learning in the specific systems it relies upon, and, as a result,a learner who is simultaneously learning abstract and specificknowledge is almost as efficient as a learner with an innate(i.e. fixed) and correct abstract theory. Hierarchical Bayesianmodels have been used before to study domain-specific abstract causal knowledge (Kemp, Goodman, & Tenenbaum,2007), and simple relational theories (Kemp et al., 2008).
Here we combine these approaches to study knowledge of
1 Cf. the “curse of dimensionality”.

Law #1:
Law #2:
Law #3:
Law #4:
Law #5:
Law #6:
Law #7:
Law #8:Law #9:Law #10:
Law #11:∀x ∀y A(x) → ¬R(y, x)∀x A(x) → Fy R(x, y)∀x F1 (x) → A(x)∀x F2 (x) → A(x)∀x ∀y R(x, y) ∨ R(y, x) ∨ x=y∀x ∀y ¬R(x, y)∀x Fy R(x, y)∀x Fy R(y, x)∀x ∀y ∀z R(x, y) ∧ R(y, z) → R(x, z)∀x ∀y A(x) → ¬R(x, y)∀x Fy ¬A(y) ∧ R(y, x)Interventions are exogenous.
Interventions have at most one child.
Feature 1 is diagnostic for interventions.
Feature 2 is diagnostic for interventions.
Dependence graph is fully connected.
Dependence graph is unconnected.
Variables have at most one child.
Variables have at most one parent.
Dependence graph is transitive.
Interventions have no children.
Variables have at most one non-intervention parent.

Figure 1: Eleven laws that can be expressed in the language for theories. The informal gloss on the right describes what each law might mean
within a theory of causality.

causality at the most abstract, domain general level.
We will also explore the possibility that learning at the abstract level in an HBM, and the blessing of abstraction, canbe substantially aided by providing appropriate low-level features in the input. Our motivation for considering this possibility is a suggestion by Carey (2009) that part of infants’core knowledge is in the form of perceptual input analyzers:
modules that perform simple transformations of raw perceptual input, making it suitable for conceptual cognition. We
hypothesize that these perceptual input analyzers do not provide abstract conceptual knowledge directly, but instead serve
to make latent abstract concepts more salient and thus morelearnable. For instance, the feeling of self-efficacy, advocated by Maine de Biran as a foundation of causality (seediscussion in Saxe & Carey, 2006), could be an analyzer
which highlights events resulting from one’s own actions,making the latent concept of intervention more salient. Altogether this suggests a novel take on nativism—a “minimal
nativism”—in which strong, but domain-general, inference
and representational resources are aided by weaker, domainspecific perceptual input analyzers.
In the following sections we first formalize aspects of CBN
within a logical language for intuitive theories. We then study
the ideal learner of causal knowledge, investigating the speedof learning at different levels of abstraction, and the effect ofperceptual input analyzers on learning speed.

Theories of causality
Causality governs the relationship between events. Formalizing this, the world consists of a collection of causal systems,
in each causal system there is a set of observable causal variables. Causal systems are observed on a set of trials—on
each trial, each causal variable has a value. (We will call anobservation of a causal variable on a particular trial an event.)
The causal Bayes nets theory of causation (Pearl, 2000) describes the structure of dependence between events, isolatinga special role for a set of interventions. CBN can be seen as acollection of assumptions about causal dependence: (CBN1)
Dependence is directed, acyclic, and can be quantified as conditional probability. (CBN2) There is independence / indirect

dependence. (CBN3) There is a preferred set of variables, the“interventions”, which are outside the system—they dependon nothing. (CBN4) Interventions influence only one variable. (CBN5) The intervention set is known for each causal
system. In addition, assumptions are often made about thefunctional form of dependence (for instance, that interventions are “arrow breaking”). For simplicity we will address
only the aspects of this theory that determine the structure of
the dependency relation and will assume (CBN1).

A language for theories of causal dependence
We wish to specify a hypothesis space of alternative theories of the dependency relation, R. This space should contain
CBN and a wide set of alternative theories, and should build
these theories by combining simple primitive units. Kemp etal. (2008) proposed a very flexible language for expressing
relational theories, which is a small extension of first-orderlogic, and used this language to predict the inductive generalization of human learners in a novel domain. We proposethat a version of this language can be used to capture domaingeneral knowledge, including (aspects of) a theory of causality.
The language we use contains logical operators: quantifiers over causal variables—“for all” (∀), “there exists” (∃),
and “there exists at most one” (F)—and logical connectives—
not (¬), and (∧), or (∨), if (←). In addition to the logicaloperators, and the causal dependence relation R(·, ·), the language contains invented predicates and observed predicates.
Invented predicates are not observable, or pre-defined, buthave a conceptual role in the theory. We restrict in this paper
to at most one invented predicate, A(·); this predicate need
not a priori relate to causality in an interesting way, but it
will play the role of defining intervention in the correct theory. Finally, the two predicates, Fi (·), are observable featuresof variables. These can be thought of as perceptual input analyzers extracting some feature of events2 , which may or may
not be useful in a theory of causality.
2 It is most realistic to think of input analyzers operating at thelevel of specific events; we idealize them as features of causal variables (i.e. types of events).

Fig. 1 gives examples of laws that can be expressed in thislanguage. These laws include those needed to capture the
CBN theory of causality, as well as a variety of plausible variants describing alternative restrictions on dependency. Within
this set of laws (CBN3) corresponds to Law #1; (CBN4) corresponds to Law #2; (CBN5) follows from Laws #3 and/or#4 when the features can be used to identify interventions;(CBN2) is the lack of Laws #5 or #9.

Theory∀x, y A(x) → ¬R(y, x)TSystem1:

Causalmodel

System2:

Rs,AsA hierarchical Bayesian model

Variables

Generating a theory A causal theory—represented in thetheory language described in the previous section—is drawn
from the prior distribution over theories, P(T ). We take P(T )to be uniform over theories (of size less than some maximum). While a representation-length prior (see Kemp et al.,2008) would naturally capture a bias for simpler theories, wechoose a uniform prior in order to focus on the dynamics oflearning driven entirely by the hierarchical setup.
Generating causal models Next a causal model is generated for each causal system s. A causal model is an instantiation of each predicate in the theory—Rs and, if it is used, As .
Following (Kemp et al., 2008), we will assume that the distribution on causal models, P(As , Rs |T ), is uniform over thoseconsistent with T —that is, the instantiations of Rs and As thatsatisfy the logical laws of T .
Generating events Each causal model in turn generates observed events (a value for each variable) for a set of trials.
The probability of generating a series of trials D = {dt } froma system with causal relation R is given by:
Z

P(D|R) =∏ P(dt |R, Θ)P(Θ|α)dΘ(1)tWhere the conditional probability tables, Θ, list the probability of each event given each set of values for its parents in R.
We make the weak assumption that each entry of Θ is drawn
independently from a symmetric-beta distribution with hyperparameter α. The integral in Eq. 1 is a product of standard
beta-binomial forms, which can be integrated analytically.

Theory induction
The ideal Bayesian learner infers a posterior belief distribution over theories from a set of observed trials across a rangeof causal systems. The posterior probability of a theory, T ,given data, D = {Ds } is given by:
P(T |D) ∝ P(D|T )P(T )(2)

1 0 1 11 0 1 1

Events

0 0 0 10 0 0 1ds,t

1 1 0 11 1 0 1

Trials t
Systems s

Trials...

To ground this language for theories into observed events ina set of causal systems, we construct a hierarchical Bayesianmodel with theories of causality at the most abstract level andevents at the most specific level (Fig. 2). We first describe thegenerative process of this model, then we describe the ideal
learner by inverting this process using Bayes’ rule.

1 0 0 1Figure 2: The hierarchical Bayesian model, and examples of theinformation at each level. The causal dependence relation R(·, ·)is shown as directed edges between variables (circles), the latentpredicate A(·) is shown as shading of the variables. Binary eventsfor each system, trial, and variable are shown as contingency tables.

Where the likelihood is given by:
P(D|T ) = ∏ P(Ds |T )s= ∏ ∑ P(Ds |A, R)P(A, R|T )
s A,R

(3)

= ∏ ∑ P(Ds |R)P(A, R|T )s A,RSystem marginals The effect of an abstract theory on
learning in a specific system, s, may be described by the posterior belief distribution over Rs . If we fix a theory, T , anduse this to provide the prior over Rs , the posterior is given by:P(Rs |T, Ds ) ∝ P(Ds |Rs )P(Rs |T )(4)

If the theory is not fixed, but is learned simultaneously withthe causal systems, we may still want to capture what has
been learned about one specific system within the hierarchicalsetup. This is given by the posterior marginal of Rs :P(Rs |D) = ∑ P(Rs |T, D)P(T |D)T= ∑ P(Rs |T, Ds )P(T |D)(5)

T

Ideal learner simulations
To investigate the dynamics of learning in the theory induction framework outlined above, we performed a series of simulation studies.
The probability landscape of this model is complex, making it difficult to accurately characterize learning at all levelsof abstraction. To ensure correct results, we chose to implement the learning model by explicit enumeration over theories and causal structures. To make this enumeration possible

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Figure 3: (a) Rank of the correct theory, mean and 10th/90th percentiles across 100 model runs. (b) Rank of the correct causal structure(mean over systems and runs), given no theory, fixed correct theory, and simultaneously learned theory. Learning abstract knowledge always
helps relative to not having a theory, and is quickly as useful as an innate, correct theory. (c) The probability of correct learning: the fractionof systems in which the correct structure has been learned (is at rank 1), and the fraction of runs in which the correct theory has been learned.
(In each run there were 50 systems, and one feature perfectly diagnostic of interventions. Hyperparameter α=0.5.)

The blessing of abstraction
Abstract knowledge acts as an inductive bias, speeding thelearning of specific causal structure. Fig. 3b shows the meanrank of the correct causal structure across systems with noabstract theory (i.e. a uniform prior over causal relations),
with innate (i.e. fixed) correct theory, and with learned theory (i.e. with the theory learned simultaneously with specific causal models). We see, as expected, that the correct
abstract theory results in quicker learning of causal struc-ture than having no theory. Comparing the learned-theory
curve to the no-theory curve, we see that abstract knowledge helps at all stages of learning, despite having to learn
it. Comparing the learned-theory curve with the innate-theory
curve shows that by around 60 samples per system the theory
learner has matched the performance of a learner endowedwith an innate, correct theory. Thus, the abstract layer ofknowledge can serve a role as inductive bias even when the
abstract knowledge itself must be learned—learning a theoryof causality is as good (from the perspective of causal model
learning) as having an innate theory of causality.
0−1

Log−posterior of corect theory

we restricted to theories which can be formed as a conjunction of at most five of the laws shown in Fig. 1, and to systems of only four variables. (Counting only theories with asatisfying causal model, there are 691 theories in the set we
considered. There are 543 possible causal structures R, and16 possible intervention sets A.)
For each run of the model we generated evidence for thelearner by first choosing one variable in each system to be
an intervention, then generating a causal model for each system (consistent with the correct, CBN, theory of causality)and data for each trial according to the generative process described above. We initially fixed the number of systems to
50, and included one feature which correlates perfectly withintervention and another which is uncorrelated with intervention; we consider the effect of varying these conditions below.
We explore the dynamics of learning by varying theamount of evidence given to the learner, as measured by the
total number of samples (i.e. trials) across all systems, witheach system given the same number of samples. The ideal
Bayesian learner is able to learn the correct theory, given sufficient evidence (Fig. 3a). This, by itself, is unsurprising—indeed, Bayesian induction is guaranteed to converge to thecorrect hypothesis in the limit of an infinite amount of evidence. It is more interesting to see that learning the correct
theory appears relatively quick in this model (being achieved
with fewer than 30 samples per system in most runs).

5 systems
10 systems
20 systems
50 systems
150 systems
500 systems

−2

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Figure 4: The posterior log-probability of the correct theory as afunction of total number of samples across systems, for different
numbers of systems. Each curve starts at 2 samples per system.
Learning is best when evidence is gathered from many systems, evenwhen only a few samples are taken in each system.

How can abstract knowledge appropriately bias specificlearning, when it must be learned itself? Comparing Fig. 3a
to Fig. 3b suggests that the correct theory is learned before

5

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Marginal probability of law

0.9
0.8

Law #1
Law #2
Law #3
Law #4
Law #5
Law #6
Law #7
Law #8Law #9Law #10
Law #11

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Figure 5: Learning curves for the eleven laws of Fig. 1.

most of the correct causal structures. In Fig. 3c we have investigated this by plotting the probability of learning (definedas the correct hypothesis being most probable), at the levels of both systems and theories. We see that learning at theabstract theory level is much faster than at the system level.
Further, the time to correct learning at the system level is almost identical for innate-theory and learned-theory, which areboth faster than no-theory. This illustrates the fact that abstract learning is not bottom-up, waiting on specific learning;
instead, learning is being carried out at all levels simultaneously, and here abstract knowledge is often learned before
specific knowledge. Note that this effect is not due to therelative size of hypothesis spaces (we consider 691 theories
and 542 specific causal structures), nor a “helpful” choice ofprior (we use a maximum entropy—uniform—prior on theories, and for the no-theory case a similar prior on specificsystems). Rather, this effect is driven by the ability of thehigher level of the model to learn from a wide range of evidence drawn from multiple systems.
To confirm that breadth of evidence is important, we consider the effect of distributing the same amount of evidence
among a different number of systems—is it better to spend effort becoming an expert in a few systems, or to be a dilettante,learning only a small amount about many systems? Fig. 4
shows the result of varying the number of systems, while
matching the total number of samples (resulting in differing
numbers of samples per system). Learning is fastest whenevidence is drawn from a broad array of causal systems, evenwhen only a few samples are observed in each system. Indeed, at one extreme learning is very slow when only fivesystems are available. At the other extreme, learning from500 systems is quick overall, and “catches up” to other conditions after only three samples per system.
Turning to the dynamics of learning for individual laws,
Fig. 5 shows the marginal probability of each of the elevenlaws in Fig. 1. Law #3, relating interventions to the ob-served predicate F1 , is learned first, but is closely followedby Law #1, which defines the main role of interventions in
CBN. Slightly later, Law #2—specifying that interventions
effect only one variable—is learned. All other laws slowly
drop off as the correct theory becomes entrenched. The gradual learning curves of Fig. 5, which are averaged over 100
runs of the model, belie the fact that learning of the laws was
actually quite abrupt in most runs. Though the exact timingof these learning events was distributed widely between runs,
the order of acquisition of the laws was quite consistent: in
two-thirds of runs Laws #1 and #3 were learned almost simultaneously, followed later by Law #2. (To be precise, in 92%
of runs Law #2 was learned last, as measured by number ofsamples required to cross probability 0.75; of these runs, Law#1 led Law #3 on 59% of runs, but the two laws were learned
within one step of each other on 74% of runs.) This observation may be significant given that cognitive development ischaracterized by wide variation in timing of acquisition, but
remarkable consistency in order of acquisition.

A minimal nativism
Thus far we have assumed that there is an observed feature
which can be used to tell when a variable is an intervention. We can imagine that this feature provides information extracted from perception of the observed events—that
is, it results from an input analyzer (Carey, 2009): an innate mechanism that performs simple transformations of perceptual evidence. A number of relatively simple input analyzers could provide features useful for identifying interventions. For instance, the feeling of self-efficacy discussed byMaine de Biran, or, more broadly, an innate agency-detector
able to identify the actions of intentional agents (see Saxe &Carey, 2006). Critically, none of these simple input analyzers
is likely to identify all interventions (or even most), and they
are likely to be mixed together with features quite un-usefulfor causal learning.
We simulated learning under several different “input analyzer” conditions varying in: the number of useful features(the remaining feature(s) were distractors), what portion ofintervention variables could be identified from the useful features, and the overlap between features. In Fig. 6 we haveplotted the marginal probability of the “intervention” portionof the correct theory—Laws #1 and #2, which govern the roleof interventions in determining causal dependency, independent of the identification of interventions. We see that learning is extremely slow when no features are available to help
identify interventions. In contrast, learning is about equallyquick in all other conditions, depending slightly on the coverage of features (the portion of interventions they identify)
but not on how this coverage is achieved (via one or multiple features). Thus, even a patchwork collection of partial
input analyzers, which pick out only a portion of intervention
variables, is sufficient to bootstrap abstract causal knowledge;
learning can be relied on to pick the useful features from the
distractors and to sort out the underlying truth that each partially represents.

1

Marginal posterior of corect abstraction

0.9
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F1 100%, F2 0%
F1 50%, F2 50%
F1 50%, F2 0%
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Figure 6: The marginal probability of the correct theory of intervention (i.e. Laws #1 and #2) given different sets of “input analyzers”:
each condition has two features which are diagnostic of intervention
variables to the extent indicated (e.g. “F1 50%” indicates that thefirst feature covers half of interventions). In the 50%/25% case thetwo features overlap, otherwise they are disjoint. Learning is difficult when no diagnostic features are present, but quite rapid under
all other conditions.

Discussion and conclusion
We have studied an ideal Bayesian learner acquiring aspectsof a domain-general intuitive theory of causality. This theory and a wide set of alternatives were represented in a “language of thought” for relational theories, based upon firstorder logic. We found that the correct theory of causality canbe learned from little evidence, often becoming available before specific causal models have been learned. This enabled
the learned abstract knowledge to act as an inductive bias
on specific causal models nearly as efficiently as an innately
specified theory. However, this “blessing of abstraction” itself relied on a set of observed event features that served tomake the latent concept of intervention more salient.
The abstractness of a theory of causality proved not to hinder learning, given a rich language of thought and a powerful inductive learning mechanism. We found that abstract
learning was fastest when evidence was drawn from a wide
variety of causal systems, even if only a small number of observations was available for each system. Because a domaingeneral theory is able to draw evidence from the widest set ofexperiences, this suggests that domain-general intuitive theories may, in some cases, be easier to learn than their domainspecific counterparts. In future work we plan to investigate
further the effects of distribution and variety of evidence.
Though we have argued that causality may be learnable,
our results should not be taken to support an entirely empiricist viewpoint. We endow our learner with a rich language
for expressing theories and a strong inductive learning mechanism. These are both significant innate structures, though
ones that may be required for many learning tasks. In addition, we have shown that the domain-general mechanisms

for learning and representation are greatly aided by a collection of domain-specific “perceptual input analyzers”. It
may be ontogenetically cheap to build innate structures thatmake some intervention events salient, but quite expensive
to build an innate abstract theory (or a comprehensive analyzer). Since a powerful learning mechanism is present in
human cognition, the most efficient route to abstract knowledge would then be by bootstrapping from these simple, nonconceptual mechanisms. Thus we are suggesting a kind ofminimal nativism: strong domain-general inference and representational resources, aided by weak domain-specific perceptual input analyzers.
Our results are purely computational, at the level of ideal
learning, but they provide a viewpoint that we believe will
be useful for empirical research in cognitive development.
When young infants behave as if they have a piece of abstract
knowledge, it is tempting to conclude that this knowledge isinnate. This tendency may misguide—we have shown that
abstract knowledge of causality, at least, can be learned soquickly that it might seem to be innate. On the other side,
where innate structure is required to explain complex cognition, it is often assumed to be abstract conceptual knowledge(Carey, 2009). This should also be approached with care—
simpler innate structures, without conceptual content, may besufficient when paired with a powerful learning mechanism.
Finally, the most obviously acquired systems of conceptual
knowledge are coherent explanations of a single domain, yet
it may often be the broader domain-general intuitive theories
which are acquired earliest and are most fundamental.

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