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Children's Imitation of Action Sequences is Influenced by Statistical Evidence and Inferred Causal Structure

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Children are ubiquitous imitators, but how do they decide which actions to imitate? One possibility is that children might learn which actions are necessary to reproduce by observing the contingencies between action sequences and outcomes across repeated observations. We define a Bayesian model that predicts that children will decide whether to imitate part or all of a sequence based on the pattern of statistical evidence. To test this prediction, we conducted an experi-ment in which preschool children watched an experimenter repeatedly perform sequences of varying actions followed by an outcome. Children's imitation of sequences that produced the outcome increased, in some cases resulting in production of shorter sequences of actions that the children had never seen performed in isolation. This behavior is consistent with our model's predictions, and suggests that children attend to statistical evidence in deciding which actions to imitate, rather than obligately imitating successful actions.
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Children’s Imitation of Action
Sequences is Influenced by Statistical Evidence and Inferred Causal Structure
Daphna Buchsbaum, Alison Gopnik, Thomas L. Griffiths
{daphnab, gopnik, tom_griffiths}@berkeley.edu
Department of Psychology, University of California, Berkeley, Berkeley, CA 94720 USA
Abstract
Children are ubiquitous imitators, but how do they decide
which actions to imitate? One possibility is that children
might learn which actions are necessary to reproduce by
observing the contingencies between action sequences and
outcomes across repeated observations. We define a Bayesian
model that predicts that children will decide whether to imitate
part or all of a sequence based on the pattern of statistical
evidence. To test this prediction, we conducted an experi-
ment in which preschool children watched an experimenter
repeatedly perform sequences of varying actions followed by
an outcome. Children’s imitation of sequences that produced
the outcome increased, in some cases resulting in production
of shorter sequences of actions that the children had never
seen performed in isolation. This behavior is consistent with
our model’s predictions, and suggests that children attend to
statistical evidence in deciding which actions to imitate, rather
than obligately imitating successful actions.
Keywords: Cognitive development; Imitation; Statistical
learning; Causal inference; Bayesian inference
Introduction
Learning the causal relationships between everyday se-
quences of actions and their outcomes is a daunting task.
How do you transform a package of bread, a jar of peanut
butter and a jar of jelly into a peanut butter and jelly sand-
wich? Do you cut the bread in half before or after you put
together the sandwich? Can you put the peanut butter on first,
or does it always have to be jelly first? In order to achieve
desired outcomes from everyday goals such as eating a
tasty sandwich to distinctive human abilities such as making
and using tools children need to solve a challenging causal
learning problem: observing that the intentional actions
of others lead to outcomes, inferring the causal relations
between those actions and outcomes, and then using that
knowledge to plan their own actions.
To learn from observation in this way, children cannot sim-
ply mimic everything they see. Instead, they must segment
actions into meaningful sequences, and determine which
actions are relevant to outcomes and why. Recent studies
of imitation in children have produced varying answers to
the question of whether children are capable of solving this
problem. While children sometimes selectively reproduce
the most obviously causally effective actions (Williamson,
Meltzoff, & Markman, 2008; Schulz, Hooppell, & Jenkins,
2008), at other times they will “overimitate”, reproducing
apparently unnecessary parts of a causal sequence (Whiten,
Custance, Gomez, Teixidor, & Bard, 1996; Lyons, Young,
& Keil, 2007), or copying an actor’s precise means, when
a more efficient action for accomplishing the same goal is
available (Meltzoff, 1995). Sometimes children may produce
both kinds of behavior in the same study. In the “rational
imitation” studies by Gergely, Bekkering, and Kiraly (2002),
children saw an experimenter activate a machine with hands
free or hands confined. Children both produced exact imita-
tions of the actor (touching their head to a machine to make it
go) and produced more obviously causally effective actions
(touching the machine with a hand), though the proportion
of such actions differed in the different intentional contexts.
We suggest that these different results reflect the multiple
sources of information that contribute to a rational statistical
inference about causally effective actions. Children need
to balance their prior knowledge about causal relations, the
new evidence that is presented to them by the adult, and
their knowledge of the adult’s intentions. Moreover, in the
case of imitation there is often no single “right answer”
to the question of what to imitate. After all, a longer
“overimitation” sequence might actually be necessary to
bring about an effect, though that might seem unlikely at
first. The imitation problem can be expressed as a problem
of Bayesian inference, with Bayes’ rule indicating how
children might combine these factors to formulate different
causal hypotheses and produce different action sequences
based on those hypotheses. It is difficult to test this idea
however, without knowing the strength of various causal
hypotheses for the children. Since previous studies involved
general folk physical and psychological knowledge (such as
removing a visibly ineffectual bolt to open a puzzle box) it is
difficult to know how strong those hypotheses would be. By
giving children statistical information supporting different
hypotheses we can normatively determine how probable dif-
ferent hypotheses should be, and then see whether children’s
imitation reflects those probabilities.
It is also independently interesting to explore the role of
statistical information in imitation. Recent studies show
that children are surprisingly sophisticated in their use of
statistical information such as conditional probabilities in
a range of domains, from phonology (Saffran, Aslin, &
Newport, 1996), to visual perception (Fiser & Aslin, 2002;
Kirkham, Slemmer, & Johnson, 2002), to word meaning (Xu
& Tenenbaum, 2007). Such information plays a particularly
important role in both action processing (Zacks et al., 2001;
Baldwin, Andersson, Saffran, & Meyer, 2008; Buchsbaum,
Griffiths, Gopnik, & Baldwin, 2009) and causal inference
(Gopnik et al., 2004; Gopnik & Schulz, 2007), and allows
adults to identify causal subsequences within continuous
streams of action (Buchsbaum et al., 2009). Varying the
probabilities of events within action sequences may thus
provide a way to vary the statistical evidence those sequences
provide in favor of different causal hypotheses.
Statistical inference might be particularly important to
Observed Action Sequence Potential Causal Sequences
ABC+ ABC, BC, C
DBC+ DBC, BC, C
Total Potential Causes ABC, DBC, BC, C
Table 1: Example demonstrations, and the associated set
of potential causal sequences. Letters represent unique
observed actions, a + indicates a causal outcome.
imitation because it could allow children to not only deter-
mine the causal relationship between action sequences and
outcomes, but to identify irrelevant actions within causally
effective sequences. Imagine that I am making a peanut
butter sandwich, and that between opening the jar, and
spreading the peanut butter, I get peanut butter on my hands,
so I wipe them on a paper towel. If this is the first time
you’ve seen me make a sandwich, you might mistakenly
think that hand-wiping is a necessary step. However, after
watching me make a sandwich a couple of times, you might
notice that while opening the jar always predicts spreading
the peanut butter, it doesn’t always predict hand-wiping, and
could infer that this step is extraneous. In most previous
work on children’s imitation of casual sequences, children
observed only a single demonstration of how to generate the
outcome (e.g. Whiten et al., 1996; Lyons et al., 2007).
In this paper, we look at whether children use statistical
evidence from repeated demonstrations to infer the correct
causal actions within a longer sequence and imitate them. We
present a Bayesian analysis of causal inference from repeated
action sequence demonstrations, followed by an experiment
investigating children’s imitative behavior and causal infer-
ences. We showed preschool children different sequences of
three actions followed by an effect, using our Bayesian model
to guide our manipulation of the probabilistic evidence, such
that the statistical relations between actions and outcomes
differed across conditions in ways that supported different
causal hypotheses. We then examine which sequences
the children produced themselves, and compare children’s
performance to our model’s predictions. We conclude by
discussing our results in the context of broader work on
imitation, and causal and intentional inference.
Bayesian Ideal Observer Model
In many real world situations, the causal structure of a
demonstrated sequence of actions is not fully observable. In
particular, which actions are causally necessary and which
are superfluous may be unclear. One way children may
overcome this difficulty is through repeated observations. By
watching someone make a sandwich or turn on a lightbulb
on multiple occasions, children can pick up on which actions
consistently predict the desired outcome, and which do not.
While it is intuitively plausible that children can use the
statistical evidence in repeated demonstrations to infer causal
structure, we would like to verify that normative inferences
from repeated observations of action sequences and their
outcomes vary in a systematic way with different patterns of
data. One way to derive what the normative distribution over
ABC
Effect
DBC BC C
ABC DBC BC
Effect
C
ABC
Effect
DBC BC C
ABC
Effect
DBC BC C
Figure 1: A subset of the hypothesis space. Each box repre-
sents a hypothesis about which action sequences are causal.
causes should be is through a Bayesian model (Gopnik et al.,
2004; Griffiths & Tenenbaum, 2005). The Bayesian formal-
ism provides a natural way to represent the roles of children’s
prior assumptions and the observed data in forming their
beliefs about which action sequences are likely to be causal.
Model Details
Given observations of several sequences of actions, we
assume that children consider all sequences and terminal
subsequences as potentially causal. These include both
sequences that generate the outcome and those that do not.
For instance, if the sequence “squeeze toy, knock on toy, pull
toy’s handle” is observed, then squeeze, followed by knock,
followed by pull handle would be one possible causal se-
quence, and knock followed by pull handle would be another.
Given all of the observed sequences, we can enumerate the
potential causes (see Table 1 for an example set of demonstra-
tions and potential causes). As in previous work on children’s
causal inference, we use a Deterministic-OR model (c.f.
Cheng, 1997; Pearl, 1996), in which any of the correct
sequences will always bring about the effect. To capture the
intuition that there may be more than one sequence of actions
that can bring about an effect, we consider all of the potential
causes (such as in Table 1), as well as all disjunctions of
these causes. The base causes, together with the disjunctions
form the space of potential hypotheses, H (see Figure 1).
The learner wants to infer the set of causes, h, given
the observed data, d, where the data are composed of an
observed sequence of actions, a, and an outcome, e. Bayes’
theorem provides a way to formalize this inference. Bayes’
theorem relates a learner’s beliefs before observing the data,
their prior p(h), to their beliefs after having observed the
data, their posterior p(h|d),
p(h|d) p(d|h)p(h), (1)
where p(d|h) is the probability of observing the data given
the hypothesis is true. For Deterministic-OR causal models,
this value is 1 if the sequence is consistent with the hypoth-
esis, and zero otherwise. For example, given the hypothesis
that squeeze is the cause, a consistent observation would be,
knock then squeeze followed by music, and an inconsistent
observation would be squeeze followed by no music. When
multiple sequences of actions and effects are observed, we
assume that these sequences are independent.
A key element in this inference is the learner’s prior
expectations, p(h). Children could have a variety of different
beliefs about the kinds of sequences that bring about effects.
For instance, they could believe that longer sequences, that
include more of the demonstrated actions, are more likely to
bring about effects. Or, they could believe that there tends to
be only one correct sequence, as opposed to many possible
sequences, that cause an effect. We capture these intuitions
with a prior that depends on two parameters, β and p, which
correspond to the learner’s expectations about the length of
causal sequences and number of ways to generate an effect.
We formalize the prior as a generative model. Hypotheses
are constructed by randomly choosing causal sequences, a.
Each sequence has a probability p
a
of being included in each
hypothesis and a probability (1 p
a
) of not being included,
p(h)
ah
p
a
a/h
(1 p
a
) (2)
where the probability of including causal sequence a is
p
a
=
1
1 +
1p
p
exp(β(|a|2))
, (3)
and |a| is the number of actions in the sequence a. Values of β
that are greater than 0 represent a belief that longer sequences
are more likely to be causes. Values of p less than 0.5 repre-
sent a belief that effects tend to have fewer causes. Together,
Equations 1, 2 and 3 provide a model of inferring hypotheses
about causes from observed sequences and their effects.
In our experiments, rather than probing children’s beliefs
directly, we allow children to play with the toy. Therefore, to
complete the model, we must specify how children choose ac-
tion sequences, a, based on their observations, d. Intuitively,
we expect that if we know the set of causes of the effect, h, we
will randomly choose one of these actions. If we were unsure
about which of several possible causes was the right one, then
we may choose any of the possible contenders, but biased to-
ward whichever one we thought was most likely. We capture
these intuitions formally by choosing an action given the ob-
served data, p(a|d), based on a sum over possible hypotheses,
p(a|d)
hH
p(a|h)p(h|d), (4)
where p(a|h) is one if a is a cause under h, and zero
otherwise, and p(h|d) is specified in Equation 1.
A Simple Modeling Example
We can now verify that the model makes distinct inferences
from repeated demonstrations. In the first example, the
demonstrated action sequences are ABC+, DBC+ as in Table
1. That is, a sequence of three actions A, B and C is followed
by an effect. Subsequently, a different sequence of three ac-
tions, D, B, and C is followed by the same effect. In the sec-
ond example, the observed sequences are ABC+, DBC. Here,
the second three-action sequence is not followed by the effect.
Observed Sequences ABC DBC BC C
ABC+, DBC+ 0.23 0.23 0.27 0.27
ABC+, DBC 1.0 0.0 0.0 0.0
Table 2: Example model results, p = 0.5 and β = 0.
Observed Sequences ABC DBC BC C
ABC+, DBC+ 0.26 0.26 0.35 0.13
ABC+, DBC 1.0 0.0 0.0 0.0
Table 3: Example model results, p = 0.1 and β = 1.0.
Using values of p = 0.5 and β = 0 results in a prior that
assigns equal probability to all possible causal hypotheses
a uniform prior. With this uniform prior, we can now find the
probability of choosing to perform each action sequence to
bring about the effect given the observed data, p(a|d), as de-
scribed in Equation 4. Our model infers that, in the first case,
all the sequences are possible causes, with BC and C being
somewhat more likely, and equally probable. Notice that the
model infers that the subsequences BC and C are the most
likely causes, even though neither was observed on its own.
The second case is quite different. Here the model sees that
DBC and its subsequences BC and C did not lead to the effect
in the second demonstration, and infers that ABC is the only
possible cause among the candidate sequences (see Table 2).
We now use values of p = 0.1 and β = 1.0 leading the
model to favor simpler hypotheses containing fewer causes,
and causes that use more of the observed demonstration.
1
This prior does not change results in the second case, where
ABC is still the only possible cause. However, in the first
case, the model now infers that the subsequence BC is the
most likely individual cause, since it is the longest observed
sequence to consistently predict the effect (see Table 3).
Model Predictions for Children’s Inferences
Our rational model makes differential predictions based
on repeated statistical evidence, and is able to infer sub-
sequences as causal without seeing them performed in
isolation. We can now use the model to help us construct
demonstration sequences that normatively predict selective
imitation in some cases, and “overimitation” in others. If
children are also making rational inferences from variations
in the action sequences they observe, then their choice of
which actions to imitate in order to bring about an effect
should similarly vary with the evidence. We test our predic-
tion that children rationally incorporate statistical evidence
into their decisions to imitate only part of an action sequence
versus the complete sequence in the following section.
Experiment
Method
Participants Participants were 81 children (M = 54
months, Range = 41 70 months, 46% female) recruited
from local preschools and a science museum. An additional
1
These parameter values qualitatively fit children’s imitative
behavior, as we discuss later in the paper.
ABC” Condition “BC” Condition “C” Condition
ABC+ ABC+ ABC+
DEC ADC ADC+
ABC+ DBC+ DBC+
EDC AEC AEC+
ABC+ EBC+ EBC+
Table 4: The demonstration sequences for ABC” , “BC”
and “C” conditions. Each child observed the experimenter
performing all 5 action sequences in their condition.
18 children were excluded from the study because of demon-
stration error (4), equipment failure (3), lack of English
(1), unavailable birth date (1), did not try toy (6), extreme
distraction (2), never performed trial termination action (1).
Stimuli There were two novel toys: a blue ball with
rubbery protuberances, and a stuffed toy with rings and tabs
attached to it. Six possible actions could be demonstrated on
each toy. Children were assigned to one of three experimental
conditions. In each condition, they saw a different pattern
of evidence involving five sequences of action and their
outcomes. Each individual action sequence was always three
actions long. In the ABC” pattern, the same sequence of
three actions (e.g. A=Knock, B=Stretch, C=Roll) is followed
by a musical effect three times, while in the “BC” pattern a
sequence composed of a different first action, followed by
the same two-action subsequence (e.g. A=Squish, B=Pull,
C=Shake and D=Flip, B=Pull, C=Shake) is followed by the
effect three times (see Table 4). In both patterns, two addi-
tional sequences that end in C and do not contain BC fail to
produce the effect. Finally, in the “C” pattern the sequences
of actions were identical to those in the “BC” pattern, but
the outcome was always positive. The number of times each
individual action is demonstrated in each sequence position is
identical in all three patterns. As we show later in the paper,
our Bayesian ideal observer model confirms that the statistical
evidence in each pattern supports different causal inferences.
Procedure The experimenter showed the child one of the
toys, and said: “This is my new toy. I know it plays music,
but I haven’t played with it yet, so I don’t know how to make
it go. I thought we could try some things to see if we can
figure out what makes it play music. The experimenter
emphasized her lack of knowledge, so that the children would
not assume she knew whether or not any of her actions were
necessary. She then demonstrated one of the three patterns
of evidence, repeating each three-action sequence (and its
outcome) twice. The experimenter named the actions (e.g.
“What if I try rolling it, and then shaking it, and then knock-
ing on it?”), acted pleasantly surprised when the toy played
music (“Yay! It played music’!’), or disappointed when it
did not (“Oh. It didn’t go”), and pointed out the outcome
(“Did you hear that song?” or “I don’t hear anything. Do
you hear anything?”). After she demonstrated all five of the
3-action sequences, she gave the child the toy and said “Now
it’s your turn! why don’t you try and make it play music”.
Throughout the experiment the music was actually triggered
Condition Triplet Double Single Other
ABC” 20 1 2 4
“BC” 10 7 0 10
“C” 8 0 8 11
Table 5: Number of children producing each sequence type
by remote activation. To keep the activation criteria uniform
across conditions, the toy always played music the first time
a child produced the final C action, regardless of the actions
preceding it, terminating the trial. Only this first sequence of
actions was used in our analysis.
Children were videotaped, and their actions from the time
they were handed the toy to trial termination were coded by
the first author, and 80% of the data was recoded by a blind
coder. Coders initially coded each individual action as one of
the six demonstrated actions, or as “novel”. These sequences
were then transferred into an “ABC” type representation, and
subsequently coded as one of four sequence types: Triplet,
Double, Single or Other (defined below). Inter-coder relia-
bility was very high, with 91% agreement on the “ABC” type
representations, and 100% agreement on sequence types.
Results and Discussion
Overall results are shown in Table 5. Children produced
significantly different types of sequences across the three
conditions, p < 0.001 (two-sided Fisher’s exact test). We
will discuss results for the ABC” and “BC” conditions first,
and then return to the “C” condition.
Effect of Statistical Evidence on Imitation In their
imitation, children could either exactly reproduce one of the
three-action sequences that had caused the toy to activate
(that is, ABC in the ABC” condition or ABC, DBC or EBC
in the “BC” condition), or they could just produce BC in
isolation. We refer to these successful three-action sequences
as “triplets”, and to the BC subsequence as a “double”.
Both a triplet and a double reflect potentially correct
hypotheses about what caused the toy to activate in both
conditions. It could be that BC by itself causes the toy to
activate in the ABC” condition and the A is superfluous,
or it could be that three actions are necessary in the “BC”
condition, but the first action can vary. In both conditions BC
is followed by the effect three times.
If children automatically encode the adult’s successful
actions as causally necessary, then they should exclusively
imitate triplets in both conditions. However, if children are
also using more complex statistical information, they should
conclude that the BC sequence by itself is more likely to be
causal in the “BC” condition than in the ABC” condition,
and that the triplet sequence is more likely to be causal in the
ABC” condition than in the “BC” condition. This is in fact
what we found the number of children producing triplets
and doubles varied by condition, p < 0.01 (two-sided Fisher’s
exact test), and differed significantly between the “ABC” and
“BC” conditions p < 0.05 (two-sided Fisher’s exact test).
Effect of Differing Causal Outcomes on Imitation The
pattern of evidence in the “BC” condition is more complex
than in the ABC” condition. This may have confused chil-
dren, leading them to produce a variety of random actions,
including BC. The “C” condition controls for this possibility.
In this condition the sequences of actions were identical to
those in the “BC” condition, but the outcome was always
positive. As we show later, our Bayesian ideal observer
model confirms that this provided statistical evidence for the
hypothesis that C alone was sufficient to produce the effect.
In all three conditions, imitation of just the final C action
in isolation was coded as a “single”. As in the ABC” and
“BC” conditions, only the subsequence BC was coded as a
double in the in the “C” condition. Also consistent with the
ABC” and “BC” conditions, in the “C” condition all five
demonstrated successful sequences (ABC, ADC, DBC, AEC
and EBC) were coded as triplets.
The “C” condition is as complex as the “BC” condition.
However in the “C” condition the final action C produced
by itself reflects a likely causal hypothesis. If children selec-
tively imitate subsequences based on the data, then children
in the “C” condition should produce C more frequently than
children in the “BC” condition, and children in the “BC” con-
dition should produce BC more frequently than children in
the “C” condition. Our results support this hypothesis. Chil-
dren in the “BC” and “C” conditions differed significantly
in the overall types of sequences they produced, p < 0.001
(two-sided Fisher’s exact test), and the number of children
producing doubles and singles in the two conditions also var-
ied significantly, p < 0.001, (two-sided Fisher’s exact test).
Performance of “Other” Actions Across all three con-
ditions, children did not just obligately imitate one of the
successful sequences or subsequences they observed they
also produced new combinations of actions. Overall, the
types of “other” sequences produced did not qualitatively
differ across conditions, and appear to be a mix of ex-
ploratory behavior and genuine errors. There was a trend
towards children in the “BC” and “C” conditions performing
more of these “Other” sequences than children in the ABC”
condition p = 0.10, (two-sided Fisher’s exact test). This dif-
ference becomes statistically significant when two children
who imitated unsuccessful triplets are excluded from the
analysis, p < 0.05, (two-sided Fisher’s exact test). This result
is compatible with findings that children tend to increase
their exploratory behavior when the correct causal structure
is more ambiguous (Schulz & Bonawitz, 2007; Schulz et al.,
2008). Finally, four children performed completely novel
actions they had never seen demonstrated. All of these
children were in the “BC” or “C” conditions, consistent with
these conditions eliciting more exploratory actions.
Model Results
Supporting our experimental results, our model makes dis-
tinct predictions in each of the three experimental conditions,
showing that the data lead to differential causal inferences.
Parameter values of p = 0.1 and β = 1.0 were chosen be-
cause they produced a qualitatively good match to children’s
performances, as shown in Figure 2. The relatively high
value for β suggests that children prefer longer (complete)
causal sequences, perhaps representing a pre-existing belief
that adults usually don’t perform extraneous actions. The
relatively low value for p suggests that children employ a
causal Occam’s razor, assuming that simpler hypotheses,
which require fewer causes to explain the data, are more
likely. Overall, these results suggest that children’s imitative
choices conform closely to normative predictions.
Finally, while children performed similarly to our model’s
predictions, there were some differences in performance
as well. Children produced more triplets than our model
predicted, especially in the ABC” condition. One reason
for this discrepancy may be that children are able to use
information about the knowledge state and intentional stance
of the demonstrator that our current model cannot take
into account. Models that can incorporate intentional and
pedagogical information, in addition to statistical evidence
are an important area of future work (Goodman, Baker, &
Tenenbaum, 2009; Bonawitz et al., 2009). We are currently
developing such a model, and exploring the role of peda-
gogical cues in children’s imitation (Buchsbaum, Gopnik,
Griffiths, & Shafto, submitted).
General Discussion
In this paper, we examined whether children are sensitive
to statistical evidence in choosing the actions they imitate.
We demonstrated that children can use statistical evidence to
decide whether to imitate a complete action sequence, or to
selectively imitate only a subsequence. In particular, children
in the ABC” condition imitated the complete sequence
ABC more often than children in the “BC” condition, while
children in the “BC” condition imitated the subsequence BC
more often than children in the ABC” condition. Children’s
performance in the “C” condition demonstrated that the
differential imitation in the ABC” and “BC” conditions
could not be explained as a result of task complexity.
The design of this experiment also eliminated other simple
explanations for these results. There were the same absolute
number of BC demonstrations followed by effects in all
three conditions, but children only produced doubles in the
second condition. Similarly, the absolute number of positive
triplet demonstrations was the same in the ABC” condition
and the “BC” condition, and was smaller than in the “C”
condition, but children produced more triplets in the first
condition than in the other two conditions. Finally, the actual
sequence of actions was the same in the “BC” and “C”
conditions but children behaved differently in the two cases.
Children appeared to selectively imitate by considering the
conditional probability of the various events and outcomes,
and formulating a set of causal hypotheses based on that data.
They then produced responses that matched the probability
distribution of the hypotheses, at least qualitatively.
It is also worth noting the information-processing com-
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Figure 2: Left: Predictions of our Bayesian model. Right: Children’s actual performance in Experiments 1 and 2.
plexity of this task. Children saw thirty similar actions and
ten outcomes in each condition, and yet they appeared to
track and use this information in deciding which actions
to produce. This is consistent with other studies in which
children and adults show surprising if implicit capacities to
track statistical regularities.
These results extend earlier findings that show children
take causal and intentional information into account appro-
priately in their imitation. They show that children also take
into account statistical information about the conditional
probability of events and do so in an at least roughly norma-
tive way. The studies also suggest a rational mechanism for
the phenomenon of “overimitation” In particular, the “triplet”
responses could be thought of as a kind of overimitation,
reproducing parts of a causal sequence that are not actually
demonstrably necessary for the effect. These results suggest
that this behavior varies depending on the statistics of the data
and the probability of various hypotheses concerning them.
Other factors may also influence the child’s judgment of
various causal hypotheses. For example, knowing that the
adult is knowledgeable about the causal system, and is taking
a “pedagogical stance” towards the evidence, may lead the
child to different causal conclusions (Bonawitz et al., 2009).
We are currently investigating the effect of pedagogical
cues on imitation of causal action sequences (Buchsbaum
et al., submitted). Similarly, seeing a repeated sequence of
actions with no obvious physical causal outcome may lead
children to suspect that the actions are intended to have a
social or psychological rather than physical effect. Both
these processes might lead to greater “overimitation” which
would nonetheless be rational.
In general however, this study shows that children are
sensitive to statistical information in determining which
sequences of actions to imitate. Along with other studies,
they support the idea that Bayesian procedures of statistical
learning, procedures that allow the construction of causal
models from statistical patterns, may play a significant role
in many important kinds of early learning.
Acknowledgments. We thank Pat Shafto and Cari Kaufman for dis-
cussions on the model design, and Kimmy Yung, Mia Krstic, and
Elon Ullman for help with data collection and coding. This mate-
rial is based upon work supported by a National Science Foundation
Graduate Research Fellowship, the McDonnell Foundation Causal
Learning Initiative and Grant FA9550-07-1-0351 from the Air Force
Office of Scientific Research.
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