^{*}

Edited by: Erika Nurmsoo, University of Kent, UK

Reviewed by: Caspar Addyman, Goldsmiths, University of London, UK; Yoshifumi Ikeda, Joetsu University of Education, Japan

*Correspondence: Nobuhiko Asakura

This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Two apparently contrasting theories have been proposed to account for the development of children's theory of mind (ToM): theory-theory and simulation theory. We present a Bayesian framework that rationally integrates both theories for false belief reasoning. This framework exploits two internal models for predicting the belief states of others: one of self and one of others. These internal models are responsible for simulation-based and theory-based reasoning, respectively. The framework further takes into account empirical studies of a developmental ToM scale (e.g., Wellman and Liu,

Inferring and understanding other people's mental states such as desires, beliefs, and intentions is crucial for our successful social interactions. This ability has been referred to as having a “theory of mind” (ToM; Premack and Woodruff,

Many studies have revealed that children come to understand other people's false beliefs at around 4 or 5 years of age; in addition, such a developmental transition appears to occur gradually (e.g., Wellman et al.,

These theories have long been regarded as contrasting conceptualizations of ToM. In recent years, however, a number of researchers have advocated hybrid theories that incorporate the essences of both theory and simulation (Nichols and Stich,

In spite of a large body of empirical findings and recent theoretical advances in ToM research, relatively few studies have proposed computational models of ToM understanding, particularly false belief understanding (O'Laughlin and Thagard,

We argue that a developmental ToM scale (Wellman and Liu,

A Bayesian network is a graphical model that provides a compact representation of the joint probability distribution for a set of random variables (Pearl,

From a theory-theory perspective, Goodman et al. (

We note that the unexpected-contents task is divided into two stages. According to Wellman and Liu (

These consecutive stages can be formalized within a Bayesian network framework as follows. Consider the process of belief updating in the first stage. The initial belief about the hidden contents of the Band-Aid box (i.e., Band-Aids) comes from observing the outside of the box. Then, the updated belief (i.e., the pig toy) arises from having visual access to the inside of the box. This in turn implies that without such visual access, the initial belief wound not be updated, but instead remain in its original state.

These causal relationships can be concisely represented with a causal graphical model or a Bayesian network, as depicted in Figure

Next, consider belief reasoning in the second stage. Given the updated belief in the first stage and available information about Peter's visual access, the child makes an inference about Peter's belief about the hidden contents of the Band-Aid box. We propose that the child makes a probabilistic inference about Peter's belief using a “theory” that represents the process of belief updating in the first stage. Specifically, we propose that at the computational level (Marr,

To construct this Bayesian network, we make two assumptions. First, we assume that the child has two theories of belief formation, one applied to her and the other to Peter, each of which can be represented as the Bayesian network in Figure

These two assumptions lead us to construct the Bayesian network depicted in Figure _{S} and _{S}, representing the binary states of visual access and belief, respectively, of self; and _{O} and _{O} of others. Given _{S} to _{O}, and the other from _{S} to _{O}. This represents their causal connections and thereby reflects the second assumption. Furthermore, this Bayesian network can be interpreted as representing the joint probability distribution over all relevant variables, implying its factorization as _{O}, _{O}, _{S}, _{S}, _{O}|_{O}, _{S}, _{O}|_{S})_{S}|_{S}, _{S})

The Bayesian network in Figure

First, we introduce notations to denote the states of a binary variable. For the states of visual access to the inside of the Band-Aid box, let _{S} and _{O} take on the value of 1 when the contents are observed and 0 otherwise. For the states of belief about the contents, let _{S} and _{O} take on the value of 1 for the pig toy (true belief) and 0 for Band-Aids (false belief).

Given the ToM network, reasoning about the belief state of others amounts to estimating the state of the variable _{O} from available information about the remaining variables. For this estimation, the Bayesian approach suggests using the predicted probability _{O}|_{S} = 1, _{O} given the observed state of _{S} and the fixed state of

Here, the summation is taken over all possible values of _{O} and _{O}. Hence, to formulate this predicted probability, it is necessary to specify three conditional probabilities on the right side of the Equation (1).

First consider _{S}|_{S} = 1, _{S} = 1), _{S} will take on the value of 1 when her belief is just updated. We assume, however, that the child may fail to maintain the updated belief owing to accidental error, or the limited capacity of her working memory. Assuming that such failure occurs with small probability δ, we set

Next, consider _{O}|_{S} = 1). This conditional probability is due to our assumption that the mental states of self have an effect on the representations of those of others. As the child hears that Peter has not observed the inside of the box, _{O} should be 0 if she correctly identifies the state of Peter's visual access. However, our assumption states that the child may mistakenly attribute her state of mind to Peter. Assuming that this happens with probability 1 − π_{V}, we set

Thus, π_{V} expresses the degree to which children can appreciate the states of others' visual access, or equivalently, the knowledge states of others.

Finally, consider _{O}|_{O}, _{S}, _{B}. This prompts us to decompose _{O}|_{O}, _{S},

This decomposition implies that to represent Peter's belief, the child uses _{O}|_{O}, _{B}, or _{O}|_{S}) with probability 1 − π_{B}. Thus, π_{B} expresses the degree to which children can attribute different beliefs to others.

Note that _{O}|_{O}, _{S}|_{S}, _{O} = 1, we set

When _{O} = 0, Peter should hold the false belief (_{O} = 0) because he observes only the outside of the Band-Aid box. However, we assume that the child takes into account the possibility that Peter expects something other than Band-Aids inside the box. This can happen because the box is a container that can hold anything smaller than its size; in fact, it held the pig toy inside in the unexpected-contents task. Assuming that the child supposes such a misconception could occur with a small probability ϵ, we set

For _{O}|_{S}), we assume that it is deterministic since its probabilistic nature has already been captured with the probability 1 − π_{B}. That is, assuming that the child's state of belief is just copied to Peter's belief, we set

By using the parameterization introduced thus far, we can derive the predicted probability _{O}|_{S} = 1, _{O} = 0|_{S} = 1,

This illustrates how theory-based and simulation-based strategies are combined to perform reasoning about the belief state of others. The first term means that the child first obtains an estimate of the state of Peter's visual access using her own state (_{O}|_{S} = 1)), then feeds the estimate into the internal model of others (_{O}|_{O}, _{S}|_{S} = 1, _{O}|_{S})). This corresponds to a simulation-based strategy. The probability π_{B} acts as a gate to select one of these strategies: the child performs theory-based reasoning with probability π_{B} and simulation-based reasoning with probability 1 − π_{B}.

Then, by setting _{O} = 0 and summing _{O} and _{S} in Equation (13), we finally obtain our model of false belief reasoning:

This is the probability that given knowledge about the true state of the world, the child estimates the belief state of Peter as a false belief. In the following, we denote this probability as π_{FB}.

Our model of false belief reasoning takes four probabilities as its parameters: δ, ϵ, π_{B}, and π_{V}. Of these, the effects of δ and ϵ are likely to be limited since they are assumed to be small, random errors. This is justified by the procedure employed in all the ToM scale studies listed above. In fact, for children's false belief responses to be scored as correct, the children were first required to respond correctly to preliminary and control questions about what is usually in a Band-Aid box (i.e., Band-Aids) and what is actually in the Band-Aid box presented (i.e., the pig toy). Thus, we can safely assume that children rarely, if ever, came up with something other than Band-Aids inside the box (i.e., small ϵ) and failed to maintain the updated belief about the contents of the box (i.e., small δ). In contrast, the remaining two probabilities, π_{B}, and π_{V}, play a dominant role in specifying the behavior of our model. Let us remember that π_{B} and π_{V} are introduced to quantify children's abilities to differentiate their mental states, beliefs, and visual access, respectively, from those of others. These abilities as well as false belief reasoning are, in fact, ToM skills that are to be assessed with the ToM scale (Wellman and Liu, _{B} corresponds to the proportion correct for the diverse-beliefs task, and π_{V} for the knowledge access task. Obviously, π_{FB} amounts to children's proportion correct for the unexpected-contents false-beliefs task.

Our model thus predicts that, when assessed with the ToM scale in terms of the proportion correct, children's false belief ability can be predicted through their abilities to understand diverse beliefs and knowledge access. Indeed, assuming that δ and ϵ are sufficiently small, we can approximate Equation (14) to obtain a simple relation: π_{FB} ≈ π_{B}π_{V}. It follows that false belief reasoning can be viewed as a multiplicative effect of understanding diverse beliefs and knowledge access. Below we will show that this simple multiplicative relation holds across a wide variety of children's ToM scale data.

We illustrated the validity of our model by fitting the full model of four parameters using a Bayesian method to the children's ToM scale data for the three above-mentioned tasks. The data for each task consist of the number of children who successfully completed the task. To fit our model to the data, we take the proportion of children who were correct on each task as their proportion correct for the task. Specifically, we assume an individual child's responses to these tasks as independent Bernoulli trials with success probabilities π_{B} for diverse beliefs, π_{V} for knowledge access, and π_{FB} for unexpected-contents false belief. This allows us to derive the joint likelihood function of the parameters π_{B}, π_{V}, δ, and ϵ (note that π_{FB} is a function of them). In addition, similar to Goodman et al. (_{FB} using Equation (14).

Figure

_{B}, π_{V}, and π_{FB}). The left part of the figure depicts the results for Western children (Australian and American); the right part for Asian/Middle Eastern children (Iranian, Chinese, and Japanese).

For three ToM tasks considered in our model, the above studies revealed different orders of difficulty between cultures. Western, English-speaking children mastered the tasks in the following order: diverse beliefs, then knowledge access, and finally false beliefs. In contrast, Asian/Middle Eastern peers reversed the order between diverse beliefs and knowledge access (Note that these orders only indicate cross-sectional ToM progressions. However, Wellman et al. (

Note that our model is in good agreement with the data from Hiller et al. (_{B}, π_{V}, and π_{FB}) and the data over all age groups was 0.98. These results indicate that our model can capture the pattern of children's ToM abilities at each stage of their development.

_{B}, π_{V}, and π_{FB}).

Finally, we assessed whether our model can apply to children with developmental delays. Peterson et al. (

_{B}, π_{V}, and π_{FB}).

We have formalized a Bayesian model of false belief reasoning that incorporates the internal models of self and others for belief formation. This model can be viewed as a version of theory-theory, explicitly representing a set of mental concepts and their interactions by a probabilistic causal model (Gopnik and Wellman,

Our model predicts children's false belief ability as a multiplicative effect of their abilities to understand diverse beliefs and knowledge access. As shown above, in terms of success probabilities for corresponding ToM scale tasks, this prediction can be concisely expressed as: π_{FB} ≈ π_{B}π_{V}. It is important to remember that the latter two probabilities are introduced into our model to represent the degree to which children can discern their mental states, beliefs, and visual access, from those of others. In effect, the larger these probabilities, the larger π_{FB}, and the stronger the tendency for children to recognize that others are not “like-me” in their mental states. Thus, our model predicts that developed false belief reasoning (i.e., larger π_{FB}) should rest predominantly on the internal model of others to employ a theory-based strategy and that conversely, undeveloped false belief reasoning (i.e., smaller π_{FB}) should be based mainly on the internal model of self to employ a simulation-based strategy. This differential weighting between the internal models of self and others enables our model to account for a wide variety of ToM scale data, capturing the variability of false belief ability observed across those behavioral studies.

Thus, our model is able to characterize children's competence in false belief reasoning at the various stages and aspects of their development. However, the model is not itself, in its current formulation, a model for ToM acquisition. Nevertheless, it provides preliminary evidence regarding how ToM development proceeds in childhood. The key point is again the multiplicative relation: π_{FB} ≈ π_{B}π_{V}. The relation states that a larger π_{FB} requires both π_{B} and π_{V} to be much larger simultaneously. Therefore, it implies that children's earlier understanding of diverse beliefs and knowledge access is a prerequisite for promoting their later false belief understanding. This naturally corresponds to a constructivist account of ToM development. Specifically, our model follows an approach of rational constructivism in cognitive development (Xu and Kushnir,

Regarding the process of ToM development, our model makes another constructivist prediction with the multiplicative relation: π_{FB} ≈ π_{B}π_{V}. A key observation is that the relation is bilinear: π_{FB} is linear in π_{B} when π_{V} is fixed and vice versa. This means that a fixed level of π_{B} or π_{V} affects the slope in the linear function of the other. Hence, provided that either π_{B} or π_{V} is fixed to a certain level and the other increases monotonically, a higher fixed level will result in a faster increase in the level of π_{FB}. This leads us to predict that children's initial level of understanding of diverse beliefs or knowledge access determines how fast their later understanding of false beliefs progresses over the course of ToM development. This prediction is qualitatively consistent with a recent microgenetic study by Rhodes and Wellman (_{B} and π_{V} are interchangeable in our model. Pursuing this idea in a future empirical study would be worthwhile to test the prediction with microgenetic methods.

We should finally note that our Bayesian model of false beliefs, however successful, is only applicable to the unexpected-contents task, and not to the change-of-location task. To formalize false belief reasoning in the latter task, we need a related but different theory, or causal structure, to represent relevant mental state concepts. Specifically, the change-of-location task involves an extra representation of other people's actions (i.e., where to look for an object). In addition, the state of their actions depends not only on that of their beliefs (where the object is located), but also on the state of their desires (whether they want the object).

Representing this causal structure as Bayesian networks, Goodman et al. (

Thus, for the change-of-location task, a similar computational explanation has been advanced to understand false belief reasoning. However, similar to most behavioral ToM studies, Goodman et al. (

NA and TI designed the study and developed the model. NA performed the model fitting to behavioral data, and prepared the manuscript. TI edited the manuscript. NA and TI discussed the results and implications of this work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was supported by a grant from Genesis Research Institute and a Grant-in-Aid for Scientific Research on Innovative Areas “Constructive Developmental Science” (25119503) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

We adopted a Bayesian method for estimating the parameters π_{B}, π_{V}, δ, and ϵ to fit Equation (14) to ToM scale data

where _{B}, π_{V}, δ, ϵ) is the likelihood function of the data and _{B}, π_{V}, δ, ϵ) is the prior probability distribution for the parameters. _{B} for diverse beliefs, π_{V} for knowledge access, and π_{FB} for unexpected-contents false beliefs. We can then represent the number of children passing each task using the sum of the Bernoulli trials with the values of 1 for success and 0 for failure. This sum has a binomial distribution, giving the likelihood function of the data for each task. The joint likelihood function of the entire data set is the product of three likelihood functions for each task and is given by:

where _{B}, _{V} and _{FB} are the numbers of children passing the corresponding tasks. The likelihood function _{B}, π_{V}, δ, ϵ) is then obtained by substituting Equation (14) into π_{FB}. For the prior probability distribution _{B}, π_{V}, δ, ϵ), we assume uniform priors on π_{B} and π_{V}, and asymmetric beta priors on δ and ϵ with a beta distribution:

As shown above, the posterior distribution is proportional to the product of the likelihood and the prior distribution. For convenience, we numerically maximized the logarithm of the product with respect to the parameters to obtain their maximum a posteriori estimates. We did this using Mathematica's built-in NMaximize function with the constraint that each parameter takes values between zero and one (i.e., all parameters should be probabilities). Substituting these estimates into Equation (14), we obtained a prediction of π_{FB}.