Intuitive Theories of Mind: A Rational Approach to False Belief
Noah D. Goodman1 , Chris L. Baker1 , Elizabeth Baraff Bonawitz1 , Vikash K. Mansinghka1
Alison Gopnik2 , Henry Wellman3 , Laura Schulz1 , Joshua B. Tenenbaum1
1
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology,
2
University of California, Berkeley, 3 University of Michigan

Abstract
We propose a rational analysis of children’s false belief
reasoning. Our analysis realizes a continuous, evidencedriven transition between two causal Bayesian models
of false belief. Both models support prediction and explanation; however, one model is less complex while
the other has greater explanatory resources. Becauseof this explanatory asymmetry, unexpected outcomes
weigh more heavily against the simpler model. We test
this account empirically by showing children the standard outcome of the false belief task and a novel “psychic” outcome. As expected, we find children whose
explanations and predictions are consistent with eachmodel, and an interaction between prediction and explanation. Critically, we find unexpected outcomes only
induce children to move from predictions consistent withthe simpler model to those consistent with the more
complex one, never the reverse.

In everyday life we often attribute unobservable mental states to one another, and use them to predict and explain each others’ actions. Indeed, reasoning about otherpeople’s mental states, such as beliefs, desires, and emotions, is one of our main preoccupations. These abilities
have been called theory of mind (Premack and Woodruff,
1978); theory of mind has become one of the most wellstudied, and contentious, areas in modern psychology.
In particular, much research has focused on the phenomenon of false belief: the ability to infer that others
hold beliefs which differ from the (perceived) state of theworld. An often-used assay of this ability is the standard
false belief task (Wimmer and Perner, 1983): the subject sees Sally place her toy in a basket, then go out toplay. In Sally’s absence her toy is moved to a box (causing her belief about the toy’s location to be false). The
subject is then asked to predict Sally’s action: “whenSally comes back in, where will she look for her toy?”
Many authors have reported that performance on thistask undergoes a developmental transition in the thirdor fourth year of life, from below-chance to above-chance
performance (see Wellman et al. (2001) for a review andmeta-analysis, though see also Onishi and Baillargeon(2005)).
It has been suggested (Carey, 1985) that domain
knowledge, such as theory of mind, takes the form of
intuitive theories, or coherent “systems of interrelated
concepts that generate predictions and explanations inparticular domains of experience” (Murphy, 1993). This
viewpoint leads to an interpretation of the false belief
transition as a revision of the child’s intuitive theory

from a copy theorist (CT) position about beliefs (i.e. beliefs are always consistent with the world) to a perspective
theorist (PT) position (i.e. beliefs are mediated by perspective, and can be false). Perhaps the most striking
comparison between these two theories is the asymmetry
in their explanatory resources: the PT theory can makefalse belief predictions and explanations while the CT
theory cannot.
Another influential thread of research has supported
the idea that human behavior is approximately rational within its natural context (Anderson, 1990). Withincognitive development both strong and weak versions of
this thesis are possible. On the strong interpretation
children respond and learn rationally throughout development; developmental stages can thus be analyzed as
individually optimal, in context, and collectively as a rational progression driven by experience. On the weakreading it is only the final, mature, state which can be
expected to be rational. The contrast between these interpretations has played out vividly in research on false
belief (cf. Leslie, 1994; Gopnik and Wellman, 1992).
Empirically, the false belief transition is slow: children do not immediately achieve false belief understanding when exposed to evidence prima facie incompatible
with the CT position (Amsterlaw and Wellman, in press;
Slaughter and Gopnik, 1996). This presents a puzzle forstrong rationality: how could it be rational to maintaina CT position about beliefs in the face of prediction failures? That is, why would one ever accept a theory withfewer explanatory resources, unless perhaps, there is adrawback to the greater flexibility of the alternative theory? Indeed, intuition suggests that greater explanatory
ability must come at the cost of greater complexity and,by Occam’s razor, it should thus require additional evidence to accept the richer theory.
However, it is difficult from an informal description
to know how explanatory resources and complexity differ between these theories, and how these factors shouldinteract with evidence. Gopnik et al. (2004) have suggested that intuitive theories may be represented as
causal Bayesian networks (Pearl, 2000); we use thisframework to specify two models1 of false belief. By
applying Bayesian methods we investigate the rationaltransition between these models, balancing explanatory
1
Our formal analysis takes place at Marr’s computational
level of modeling (Marr, 1982), that is, we describe the competencies, but not the algorithms (or processes), of cognition.

resources against complexity, and illuminate the above
revision puzzle. To probe these ideas experimentally weinvestigate children’s predictions and explanations, incases when these predictions succeed and when they fail:the false belief task with the standard outcome (surprising to CTs), and a novel “psychic” outcome (surprising
to PTs). We present only the apparatus necessary fora first investigation, leaving important elaborations for
future work.

Formal Models
In the standard false belief task, described earlier, thestory begins with Sally putting her toy in the basket.
As the story continues there are only three (observable)variables that have multiple outcomes: the final positionof the toy, Sally’s visual access to the final position (i.e.
whether the door of the basket and box are open), andSally’s action upon re-entering the room. Thus we havethe variables World, Visual Access, and Action available
to our models (see Table 1 for descriptions). In addition, there are two unobservable mental state variables:
Sally’s belief about the location of her toy, Belief, and
her Desire. We simplify the, presumably sophisticated,
sub-theory of goals and desires (see Baker et al., in press)by collapsing desires into one variable, which indicates
whether Sally’s primary desire is her toy. (Formally, we
marginalize out all other variables in this sub-theory.)
(a)

ε

W

B
V

D
A

(b)

W
V

γ

ε

B

D
A

Figure 1: The dependency graphs of our Bayesian Network
Models: (a) CT model, (b) PT model. Variables abbreviatedby their first letter (see Table 1).

To specify the relationships between these variables
we fix their joint distribution by giving a causal Bayesian
network. The pattern of conditional dependencies, givenby the directed graphs in Fig. 1, codifies the intuition
that action is determined by beliefs and desires, and that
belief is affected by the state of the world. In the PT
model belief also depends on access2 .
The conditional dependencies are parameterized by
the conditional probabilities given in Table 1. The conditional probability table for action describes a simple
case of the rational agent assumption: a person will act
rationally, given her beliefs, to achieve her desires. In
this case, if Sally wants her toy she will go to the location she believes it to be in, otherwise she goes to either
location with equal probability (surely a simplification,
but sufficient for present purposes). The variable Desire has prior probability 1 − ε, which will be large for2This is a simplification: we model how belief content depends on access, but it is likely that access mediates knowledge (vs. ignorance) even in the earlier theory.

desirable objects (such as a toy).
For the CT model, Belief is constrained to equalWorld. This is also true for the PT model when Visual Access is present, but without access Sally maintains her original belief, Belief = 0, with probability
1 − γ. The parameter γ represents all the reasons, outside of the story, that Sally might change her mind: hersister might tell her the toy has moved, she may have
E.S.P., she may forget that she actually left her toy inthe basket....
We assume asymmetric-beta priors on ε and γ. In
the example simulations described below (Figures 2 and3) the hyper-parameters were set to β(1, 10) for ε, indicating that Sally probably wants her toy, and β(1, 5)
for γ, indicating that she is unlikely to change her belief(lacking access). The relative magnitude of the two parameters determines whether it is more likely that Sally
wants something other than her toy, or that she changes
her belief – we have chosen the latter (because standard
false belief tasks emphasize that Sally wants her toy).
Otherwise, the qualitative results described below arequite insensitive to the values of these parameters.

Prediction
Having represented our models as probability distributions, rational predictive use is now prescribed by thealgebra of probabilities: conditioned on observation ofsome subset of the variables, a posterior distribution isdetermined that predicts values for the remaining variables. These models are causal theories: they also support predictions of the outcome of interventions, via thecausal do operator.)
Take the example in which Sally’s toy has been moved,
but Sally doesn’t have visual access to this new location (see schematic Fig. 2(a)). There are two possible
outcomes: Sally may look in the original location (thebasket), or the new location (the box). We may predict the probability of each outcome by marginalizing
the unobserved variables. Fig. 2(b) shows that the twomodels make opposite predictions. We see that the CT
model “fails” the false belief test by predicting that Sally
will look in the new (true) location, while the PT model“passes” by predicting the original location. The surprising outcome cases differ for the two models (looking
in the original location for CT, looking in the new location for PT). Note that while the surprising outcome is
not impossible in either model, it is far less likely in the
CT model (as evident from Fig. 2(b)). That is, there isan explanatory asymmetry: prima facie equivalent unexpected outcomes weigh more heavily against the CT
model than the PT model.

Theory Revision
Strong rationality requires an agent to balance the available intuitive theories against each other. How should atheory-user combine, or select, possible theories of a domain, given the body of her experience? Fortunately, thealgebra of Bayesian probability continues to prescribe rational use when there are competing models: the degreeof belief in each model is its posterior probability given

Variable
World (W )Access (V )Action (A)Belief (B)Desire (D)Description
Location of the toy.
Could Sally see the toy moved?
Where Sally looks for her toy.
Where Sally thinks the toy is.
Sally’s primary desire.

P (A = 1|B, D)
0
1
0.5
0.5

B
0
1
0
1

D
1
1
0
0

PCT (B = 1|W )0
1States0: Original location, 1: New location.
0: No, 1: Yes.
0: Original location, 1: New location.
0: Original location, 1: New location.
1: To find the toy, 0: Anything else.
PPT (B = 1|W, V )
0
1
γ
γ

W
0
1

W
0
1
0
1

V
1
1
0
0

Table 1: The random variables and probability distribution tables for our models.

Psychic outcome

Standard outcome(a)Original
location(b)
Original
locationNew
locationNew
location

InitialFinal

Original
location1
Protagonist
looks

Original
location

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

1
Protagonist
looks

(c)
P(A=0|W,V)

CT

PT

P(A=1|W,V)

1

0.8

0.8

0.6

0.6
0.4

New
locationNew
location

0.4
0.2

0.2

Initial

Final

0

0

CT

PT

(d)

P(D,B|W,V,A)
D:0
B:1

P(D,B,Eγ|W,V,A)
D:1
B:0
Eγ:0
D:0 D:0
B:0 B:1
Eγ:0 E :1
γ

CT

PT

P(D,B|W,V,A)

P(D,B,Eγ|W,V,A)

D:1
B:1

D:0
B:1
CT

D:1
B:1
Eγ:1

2.5

Surprise of posterior mode
D

2
1.5
1
0.5
0
2

CT

PT

D

B

PT

Surprise of posterior mode
Eγ

1.5
D:0
B:0 D:0
Eγ:0 B:1
Eγ:1

Eγ

B

B

1
0.5
0

D

B

D

CT

PT

Figure 2: Comparing the Models: (a) Example situations, in both cases W =1, V =0, for the Standard outcome A=0, for thePsychic outcome A=1. (b) The predicted probability of each outcome. (c) Posterior probability of configurations of hidden
variables, after observing the outcome. This indicates degree of belief in the corresponding complete explanation. (d) Surprisevalue of each variable in the modal configuration (computed as surprisal with respect to the predictive posterior).

previous experience. We may then write down a belief weight comparing belief in the PT model to the CT
model:
WPT/CT = − log(P (PT|X)/P (CT|X)),(1)

where X represents experience in previous false belief
settings. When WPT/CT is strongly negative the contribution from PT is negligible, and the agent behavesas though it is a pure CT. If evidence accumulates andshifts WPT/CT to be strongly positive the agent behaves
as a PT. In Fig. 3 we plot WPT/CT evaluated on accumulating “epochs” of experience. Each epoch consists oftrials with (W, V, A) observed, but (D, B) unobserved.
The trials in each (identical) epoch encode the assumptions that visual access is usually available, and that,in instances without access, the protagonist often has
a correct belief anyway (e.g. to a child, his parents often appear to have preternatural knowledge). (Specif-

ically, each epoch is twenty (W =1,V =1,A=1) trials,
six (W =1,V =0,A=1) trials, and one (W =1,V =0,A=0)
trial.) The expected transition from CT to PT does occur under these assumptions. Since this rational revision
depends on the particular character and statistics of experience, a developmental account is incomplete without
empirical research on the evidence available to children
in everyday life.
How can we understand the delayed confirmation of
the PT model? First, in the initial epoch, the CT modelis preferred due to the Bayesian Occam’s razor effect(Jefferys and Berger, 1992): the PT model has additional
complexity (the free parameter γ), which is penalized via
the posterior probability. However, the data themselves
are more likely under the PT model – because some ofthe data represent genuine false belief situations. As
data accumulates the weight of this explanatory advantage eventually overcomes complexity and the PT model704#4



























EPOCH

Figure 3: The log-posterior odds ratio over data epochs,
showing the false belief transition from CT to PT. (Parameters integrated numerically by grid approximation.)

becomes favored.
When WPT/CT is close to threshold, inferences will bemixed and far more sensitive to evidence3 . In particular,
any effect of the explanatory asymmetry noted above
should be particularly prevalent in this period. When far
from threshold, predictions in the situation of Fig. 2(a)will be largely consistent, but close to threshold they
may become mixed, and such variability will be greater
below threshold than above.

Explanation
As the above discussion emphasizes, an important function of intuitive theories is to explain observations by
hypothesizing states of unobserved variables. To model
this explanatory competence we first recast our models
into a deterministic explicit-noise form, by introducing
additional variables, in order that observations will follow necessarily from unobserved variables. Explanation
can then be described as inference of a complete explanation – a setting of all variables – and communicationof a portion of this complete explanation, the explanans.
Our PT model becomes deterministic if we introduce
an External Information (or ‘alternate access’) variable,
Eγ (with prior probability γ), to explicitly represent
events which cause changes in belief (in the absence ofaccess). (For completeness, an additional variable that
determines the object of alternate desires could be included; for technical reasons this variable has minimal
effect, and has been omitted for clarity.) Explanatory
inference is now dictated by the posterior distribution,
conditioned on observations: the degree of belief in eachcomplete explanation is given by its posterior probability (e.g. Fig. 2(c)). (See Halpern and Pearl (2001) for a
related approach.) Once a complete explanation is chosen, a partial account of the explanans can be given byappealing to the principle of surprise: a good explanans
will address the ways in which an explanation is surprising 4 .
We can now characterize the explanatory asymmetry
between the two models in more detail. Comparing thedependency structure of the two models, we see that ac3
One expects the details of a process-level account to be
especially critical near the threshold.
4
Surprise may be formalized by the information-theoretic
surprisal of a value with respect to some reference distribution, such as the predictive posterior.

cess, alternate access (including external information),and belief (independent from the world) are causally relevant variables only for the PT model. We thus expect
PT-theorists to appeal to belief and access more thanCT-theorists, while CT-theorists will appeal primarily
to desire. Indeed, to explain a surprising experience theCT model can only infer an alternate desire – Sally went
back to the basket because she wanted the basket (notthe toy). The PT model has additional explanatory resources in the sense that it can also appeal to access, particularly alternate forms of access, to explain unexpected
outcomes – Sally went to the box because someone told
her that her toy was there. Fig. 2(d) shows that thesealternate desire and alternate access variables are indeed
the surprising aspects of the most relevant complete explanations. From the principle of surprise we may thenpredict an interaction between theory and explanation
type: in the surprising outcome cases CT-theorists willappeal to alternate desires, while PT-theorists will appeal to alternate access.

Children’s Predictions and Explanations
If children are using intuitive theories of mind, as described here, then several model predictions should hold.
First, there will be an evidence-driven false belief transition, and there will be a group of children near thethreshold of this transition who exhibit decreased coherence in prediction and explanation. Because of the explanatory asymmetry between models, we predict thatsurprising outcomes will have greater weight before thetransition than after. In particular, we expect some children who begin with CT predictions to switch to PT predictions when they encounter surprising evidence, but
few children to switch from PT predictions to CT predictions when they encounter surprising evidence, andfew children in any condition to switch when given consistent evidence.
Among children who are relatively far from threshold,and thus provide consistent predictions, there should be
an interaction between prediction type and explanations:PT-predictors should generate more belief and access responses, while CT-predictors should generate more desire responses. Accordingly, we investigated the predictions and explanations that children generated when presented with the two possible outcomes to the standard
false belief story. This is an extension to the predictionexplanation paradigm used by several authors to investigate false belief (e.g. Bartsch and Wellman, 1989).
Participants Forty-ninechildren(R=3;0-4;11,
M=3;11) were tested in a quiet corner of an interactive
science exhibit at a local museum. Parents were visible
to the child, but were instructed not to interact withthe child during the study.
Materials and Procedure Three picture books werecreated. The first book presented a guessing game unrelated to the false-belief task and was used to familiarize the child with the experimenter and with generating
guesses. The other two books, Standard and Psychic, followed the standard Sally-Anne style narrative, with car-(M=3;7) to those of the Mixed-predictors (M=3;10) revealed that the Mixed-predictors were significantly older
than the CT-predictors (p < 0.05, t = 2.06). This suggests that this pattern of mixed predictions is indicativeof children who are close to the false belief transition.
Explanations were coded for the mention and value,
if any, of each variable as in Table 1. (For the remainder, we have combined External Information with Access for clarity.) For example, one (PT-predictor) childexplained the Psychic outcome by “I think he heard his
sister going over there,” and this was coded as Access=1.
Another (CT-predictor) child explained the Standard
outcome by “well, that’s where she wants to look,” which
was coded Desire=0. Responses were scored by twocoders, one who was blind to the group type for eachchild and to the formal model; inconsistencies were resolved by discussion. There were no significant differences between groups in references to observed variables(Initial World, Final World, Action).
(a)

(b)

100
80
60
40
20
0

Alternate Desire
Alternate AccessStandard

20

Psychic

15
10

PT
CT
Mi
x

PT
CT
Mi
x

5

PT
CT
Mi
x

toon pictures depicting the events throughout the book.
These stories are equivalent to the situations of Fig. 2(a).
Standard outcome book: Sally was shown hiding her
teddy-bear in a basket before going outside. Children
were asked to point out where the teddy-bear was being hidden. Then a mischievous character, Alex, wasshown moving the teddy-bear from the basket to thebox while Sally was away. As a memory check children
were asked where the teddy-bear was moved. On thethird page, Sally starts to come back into the room, andthe children were asked, “Here comes Sally. Where do
you think Sally is first going to go to get her toy?” Afterthe children responded, the next scene depicted Sally going to the basket to get her toy. The children were thenprompted with, “Sally went to look for her bear in thebasket. Sally’s bear is really in the box. But Sally is
looking for it in the basket! Why is she looking there?”
The children were given the chance to respond. If they
were unable to provide an explanation or provided uninformative information, the experimenter repeated, “Yes,but she’s looking for it way over here. What happened?”
Psychic outcome book: The procedure and dialog were
essentially identical to the Standard outcome book, except the main character searched for his missing item inthe location to which the item was moved, not where it
was left. There were also superficial differences involving
different characters (Billy & Anne), different objects (acookie), and different locations (a drawer and a cabinet).
Predictions and explanations were elicited as before. The
order of the two test books was counter-balanced.
Results and Discussion Four children failed thememory test, and were excluded from further analysis.
Based on the childrens responses to the prediction question in each book, we found three groups of children: 15
children who predicted the new location in both books(CT-predictors), 20 children who predicted the original location in both books (PT-predictors), and 10 children who changed their prediction based on the surprising evidence (Mixed-predictors). Critically, and as predicted by the explanatory asymmetry between the models, children only switched predictions after surprisingevidence, and no children moved from the PT-consistentprediction to the CT-consistent prediction. Thus, ofthe children who initially made a CT-prediction, significantly more of those who received surprising evidence (for them the standard outcome) switched thanof those who received confirming evidence (p < 0.01,
χ2 = 7.84), and of the children who received surprising evidence significantly more who initially made CTpredictions switched than of those who initially madePT-predictions (p < 0.01, χ2 = 8.60). This order effect cannot be explained as a simple response to prediction failure, because the PT-predictors showed noincreased tendency to switch when they received surprising evidence (for them the Psychic condition). One
might worry that only the younger children were flustered enough by a wrong prediction to begin random
guessing. This would imply that older children shouldbe less likely to switch predictions than younger children. In fact, comparing the ages of the CT-predictors

BeliefAccess Desire

0

PT

CT

Figure 4: Summary of explanation data: (a) Portion of responses for each prediction group and condition referencing
Belief, Access, and Desire. (b) Portion of responses asserting
alternate values of Desire and Access, by prediction group.

As predicted, more CT-predictors than PT-predictors
gave Desire explanations (p < 0.025, χ2 = 5.60, acrossconditions), and more PT-predictors than CT-predictors
gave Belief or Access explanations (p < 0.025, χ2 = 6.60,
across conditions). The Mixed-predictors gave explanations that were quite similar to the CT-predictors: theirreferences to both Desire and Belief+Access were significantly different than the PT-predictors (both p < 0.01
by χ2 ), but not the CT-predictors. Children’s explanations are summarized in Fig. 4.
There is no way to know the chance level for explanations. However, consider that for an explanation to be
coded as referring to alternate access or alternate desires,children had to spontaneously invent and report details
outside the story (e.g. “he heard his sister move it”, or“she wanted to move the basket”). It is thus suggestivethat three of twenty PT-predictors gave alternate accessexplanations, and two of fifteen CT-predictors gave alternate desire explanations, in the respective surprisingoutcome conditions. Further, as predicted by our formal
analysis, there is a significant interaction between prediction group (CT vs. PT) and type of the alternate explanations that were offered (desire vs. access) (p < 0.05 by
2×2 mixed ANOVA, F (1, 136) = 5.04). The interaction
remains significant when restricting to the surprisingoutcome conditions (p < 0.05 by 2×2 mixed ANOVA,F (1, 66) = 5.31). Interestingly, the Mixed-predictors offer significantly more alternate explanations (access ordesire) than the non-mixed (PT and CT) predictors(p < 0.025, χ2 = 5.02).

Conclusion
The history of developmental psychology has been filledwith tension between the view that children are incomplete minds biding their time until full maturation, andthe view that they are rational agents bootstrapping
their way to an understanding of the world. The notion that children are strongly rational is alluring, as it
would provide a uniform principle from which to understand development.
We have outlined a computational account of theory ofmind as applied to the false belief task. This framework
realizes false belief reasoning as rational use and revisionof intuitive theory. Few formal models have been previously presented to account for false belief, and, to thebest of our knowledge, none of these other models gives
a strongly rational account. The CRIBB model of Wahl
and Spada (2000), for instance, approaches failures offalse belief as the result of limited processing capability.
Two of the primary advantages of formal models havebeen illustrated here. First, added precision can illuminate theoretical problems that resist simple solution.
Indeed, our computational account sheds some light on
the puzzle of rational theory revision by bringing thetools of Bayesian analysis to bear on the tradeoff between explanatory resources and complexity. Second, amodel can suggest novel experimental avenues. Consideration of the formal structure of our models, especiallythe explanatory asymmetry between them, suggested designing an outcome condition which would be surprising
to children who passed the false belief test – the novel
Psychic outcome condition of our experiment. This condition then provided the crucial contrast needed to understand the interaction between theory and explanation, and to detect the outcome-order effect in predictions. These in turn suggest further experimental andtheoretical avenues, such as a training study to test oursuggestion that certain mixed predictions are a signatureof children very near to the false belief transition.
The present account of the false belief transition is incomplete in important ways. After all, our agent had
only to choose the best of two known models. This begs
an understanding of the dynamics of rational revisionnear threshold and when the space of possible models
is far larger. Further, a single formal model ought ultimately to be applicable to many false belief tasks, andto reasoning about mental states more generally. Several
components seem necessary to extend a particular theory of mind into such a framework theory: a richer representation for the propositional content and attitudes inthese tasks, extension of the implicit quantifier over trials
to one over situations and people, and a broader view ofthe probability distributions relating mental state variables. Each of these is an important direction for future
research.

Acknowledgments
Thanks to the Boston Museum of Science, participants of
the McDonnell Workshops (2005), Rebecca Saxe, Tania Lombrozo, and Tamar Kushnir. This research was supported by
the James S. McDonnell Foundation Causal Learning Collaborative Initiative.

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