Two agents A and B play the JA-NG as detailed here. Specific variables are introduced in the following subsection.

The JA-NG is a procedural description of the interaction between two agents and their learning process through the sharing of semiotic knowledge between them based on joint attention.

We first define the variables related to the JA-NG and assume a conditional dependency between the variables by defining the Inter-PGM (Figure 2). Table 1 is an explanation of the variables in the Inter-PGM. An Inter-PGM is a general form of PGM that models symbol emergence using the JA-NG.

The probability variables related to the JA-NG can be described using a probabilistic graphical model, as shown in Figure 2.

The generative process of the Inter-PGM is as follows:

where xn* represents the observed information, cn* represents the category to which xn* is classified, that is, perceptual state, and sn* represents the sign of xn*, and *∈{A, B}.

A PGM can be decomposed into two parts corresponding to the two agents using the Neuro-SERKET framework (Taniguchi et al., 2020) in the inference process. found that a certain type of language game can be regarded as a decentralized inference process for an Inter-PGM, and Taniguchi et al. (2023) formulated this idea as the MHNG.

The MH naming game is a special case of the JA-NG (Taniguchi et al., 2023). The JA-NG becomes the MHNG upon satisfying the following conditions:

It is theoretically guaranteed that the MHNG is an approximate decentralized Bayesian inference of shared representations, that is, P({sn}n=1,…,N∣{xnA,xnB}n=1,…,N) and the internal representations and the knowledge of each agent. Specifically, internal representations are characterized by the local parameters cn*, while knowledge is defined by the global parameters Θ^* and Φ^* in Figure 2. These global parameters represent the relationship between observations and internal representations, and the relationship between names and internal representations, respectively. For more details, please refer to the original article (Taniguchi et al., 2023).

We used Inter-GM, which was tailored to fit the observations, that is, the color information used in our experiment. Hagiwara et al. (2019, 2022) proposed inter-DM and inter-MDM models in which agents observed bag-of-features representations, that is, histograms. They formed individual categories using a Dirichlet mixture and shared signs linked to the formed categories through communication. Inter-GM is a modified version of inter-DM in which the part that formed categories using a Dirichlet mixture is replaced by a Gaussian mixture for categorizing multidimensional continuous real-valued vectors.

The Inter-GM generative process is as follows:

Cat(*) is the categorical distribution, N(*) is the Gaussian distribution, W(*) is the Wishart distribution, and Dir(*) is the Dirichlet distribution. The parameters for the Gaussian mixture model (GMM) {μk*,Λk*}k=1,…,K correspond to Φ^* and {θl*}l=1,…,L corresponds to Θ^* in Inter-PGM (Figure 2) respectively.

In the MHNG, after observing (or sampling) sn*, the probabilistic variables for each agent become independent, and the parameters for each agent can be inferred using ordinal approximate Bayesian inference schemes. We applied Gibbs sampling, a widely used Markov chain Monte Carlo approximate Bayesian inference procedure (Bishop and Nasrabadi, 2006), to sample the parameters μk*, Λk*, cn*, and Θ^*.

In the MHNG, the sign s_n is inferred by agents A and B through an alternative sampling of the sign s_n from each other, and acceptance based on the acceptance probability of the MH algorithm rnMH=min(1,P(cnLi∣ΘLi,sn⋆)P(cnLi∣ΘLi,snLi)) for the sign of the other agent where ΘLi={θlLi}l=1,…,L is inferred using cnLi and sn⋆.

The acceptance probability estimated from the categorization results (see Figure 3) and the actual acceptance/rejection decisions were recorded to investigate whether humans accept the proposals of their opponents based on the MH acceptance probability. The parameters Θ^* and Φ^* are inferred through Gibbs sampling using the categorization {cn*}n=1,…,N provided by the participants, along with their names sn* and original observations xn*. The MH acceptance probability rnMH is then calculated, where sn⋆ denotes the proposal of the opponent.

To investigate whether the acceptance of the speaker's proposals by a listener aligns with the acceptance probability calculated by the MH algorithm rnMH, we performed a communication experiment with human participants. Instead of the computational experiment described in Hagiwara et al. (2019), we performed a communication experiment with human participants that followed a methodology similar to that of experimental semiotics.

The experiment was conducted in pairs, referred to as participants A and B. Each pair followed the procedure outlined in Figure 3 and used separate personal computers (PCs). Participants were in different rooms and were not permitted to communicate directly using any alternative communication media.

Figure 4 shows the user interface of the experimental application. (1) in Figure 4 shows the category classification screen that the participants first encountered, (2) shows the screen for the name, and (3) shows the screen for the listener. The procedure is detailed below.

Before starting the communication, each participant was instructed to classify 15 images into categories labeled A–E (initialization).

During the experiment, the participants repeated steps 2–4 15 times for each data sample, and the process was done three times. Therefore, each participant made 45 acceptance or rejection decisions per dataset.

The communication process involved proposing and accepting/rejecting names in steps 1 and 2. Each communication was completed when step 2 ended and the results were recorded each time. Participants could modify their classification results whenever desired; however, a prompt appeared if they attempted to alter the result after accepting/rejecting the proposal of their partner when playing the role of the listener. The participant pairs were housed in separate rooms, and the classification and communication were performed on PCs using a Python application, which communicated with the other PCs. The PCs used were 13-inch MacBooks. The brightness of the PCs was automatically adjusted to account for the possibility of different ambient lighting in each room. The images were presented in random order because the same images were used even after switching roles in step 3. Figure 5 shows a photograph of an actual experiment. Participants were given the following instructions:

We used the Inter-GM described in the Preliminaries section to analyze the behavioral data and predict the acceptance rate of the participants.

The hyperparameters used for the Inter-GM were α = (0.1, 0.1, 0.1, 0.1, 0.1)^T, β = 1.0, m = (50, 0, 0)^T, W-1=(200000200000200), π = (1/5, 1/5, 1/5, 1/5, 1/5)^T, in an empirical manner.

For the experiment, 20 participants were recruited, forming 10 pairs. The female-to-male ratio was 6:14, and the minimum and maximum ages were 21 and 59, respectively. As the experiment used colors, the participants were verbally asked whether they were colorblind to ensure that colorblind participants were not included in the experiment. The study was initiated on 27 July 2022, and the recruitment of participants commenced on 15 December 2022. The experiment took place from 20 December 2022 to 25 January 2023. Throughout this period, the authors created and maintained the experimental data of the participants and correspondence tables. It is important to note that the correspondence tables were not accessed or used during the data analysis stage.

This study was approved by the Research Ethics Committee of Ritsumeikan University under approval number BKC-LSMH-2022-012. All the participants provided informed consent before participation.

To generate color images as stimuli, the CIE-L^*U^*V^* color space, which accurately represents the psychological distance perceived by humans, was used (Steels et al., 2005). In the CIE-L^*U^*V^* color space, L^* represents brightness and U^*V^* represents hue. The details of the color images were as follows: (1) Pillow (PIL), a Python image processing library, was used to create images of colored circles¹. (2) L^*, U^*, and V^* were sampled from three three-dimensional Gaussian distributions. (3) Two datasets, hard and easy, were prepared to observe the differences in communication according to difficulty levels: Dataset 1 was difficult to classify, and Dataset 2 was easy to classify. (4) The same images were shown to both participants and each dataset contained 15 images. (5) The Gaussian distribution to sample from was determined using a uniform distribution.

Table 2 lists the parameters for each Gaussian distribution. Each data point in the three-dimensional CIE-L^*U^*V^* color space was generated from a three-dimensional Gaussian distribution. Figure 6 shows images of Dataset 1 (hard), and Dataset 2 (easy).

We investigated whether the decisions of people are affected by their acceptance probability using the acceptance probability based on the MH algorithm, although the decision does not completely comply with the theory. To investigate whether humans use the MH-based acceptance probability to a certain extent, that is, whether the actual acceptance probability correlates with the MH-based acceptance probability, we defined a biased Bernoulli distribution, Bern(zn∣rn=arnMH+b). The Bernoulli distribution, Bern(z∣r), samples one with probability r and zero with probability 1−r. The weight parameter a, indicating the extent to which the inferred acceptance probability is used, and bias parameter b, indicating the degree to which acceptance occurs unconditionally, were used and estimated. If a = 1 and b = 0, the distribution becomes the original MH-based acceptance probability distribution, Bern(zn∣rn=rnMH). Specifically, variable z_n represents whether the participant accepted the given name, taking the value of 1 if accepted and 0 if rejected. The acceptance probability of a participant is denoted by r_n.

The linear transformation was introduced to account for external social and cognitive factors that might affect acceptance probability. While the MHNG theory focuses on perceptual states and their relationship to signs, as shown in Figure 1, it does not consider other social or cognitive influences on acceptance probability. Factors such as respect for a counterpart or an authority gradient (Gluyas, 2015) could make it easier to accept another's proposal. Notably, to reduce the influence of these factors during the experiment, we conducted sessions with participants in separate rooms, ensuring that they could not see each other.

We tested the estimated parameter a, which model the relationship between the actual acceptance probability and MH-based acceptance probability rnMH. For acceptance and rejection, we assumed 1 and 0, respectively. Instead of calculating the correlation between the acceptance/rejection decision and r_n, we used a conditional Bernoulli distribution.

Parameters a and b were determined using the maximum likelihood estimation. The maximum log-likelihood estimation of parameters a and b was performed using gradient descent. The original likelihood function is defined as

To avoid the Bern parameter from going outside the domain, a and b were bounded to 0 ≤ b and a+b ≤ 1, respectively.

A hypothesis test was performed to test the statistical significance of the association between the rnMH score and acceptance decisions made by human participants.

The null hypothesis H₀ and alternative hypothesis H₁ are as follows:

The test statistic is the coefficient of a (bounded) linear function that parameterizes the Bernoulli distribution and the acceptance probability as output. The test statistic was set as the coefficient of the regression fitted to the observed data â.

To estimate the sampling distribution of the test statistic, we used a randomized approach in which we randomly generated Bernoulli random variables with a fixed parameter and then fitted a linear model to obtain the coefficient a (i.e., the test statistic) from the null hypothesis². The acceptance and rejection decisions were randomly sampled from the distribution by assuming H₀, that is, zn~Bern(z∣b̄). The null distribution of the test statistics was estimated and the p-values were empirically calculated. By performing this 1,000 times, we obtained an estimate of the sampling distribution as a histogram, which we used to compute the p-value as the tail probability. b̄ was determined from the behavior of all subjects using maximum likelihood estimation.

By assuming that the acceptance events occur with probability r_n, we can compute the likelihood by fitting them to the Bernoulli distribution and multiplying them by the total number of given names N; that is, L=∏n=1NBern(zn∣rn=arnMH+b)

We performed sampling using Bern(z̄∣b̄) to obtain lists of test statistics a and created their cumulative distribution functions to conduct a statistical test. The following steps describe the process of obtaining the list of test statistics a: From the experimental results, we calculated the acceptance rate b̄=1N∑n=1Nzn for all participants or target participants across all trials. We sampled the acceptance or rejection of each round from the Bernoulli distribution Bern(z̄∣b̄) with the parameter b̄ determined in the previous step, that is, z̄n~Bern(z̄∣b̄) (n∈1,…,N). Parameter a was estimated using the maximum likelihood estimator for each sampling result and was added to the list of statistical quantities. This procedure was performed 1, 000 times and the sample distributions of a were obtained.

We computed the cumulative distribution function Pa′(a)=1L∑l=1Lf(al,a) from a list of obtained statistical values a represented as a₁, a₂, …, a_L, where L = 1, 000. Similarly, we computed the cumulative distribution function Pb′(b)=1L∑l=1Lf(bl,b) from a list of statistical values b, represented by b₁, b₂, …, b_L. Here,

where f(x, y) represents a function that returns 1 if x exceeds or is equal to y, and 0 if x is below y. Because r_MH can be used if it is significantly greater than 0, a one-sided test was performed.

In Test 2, we tested whether the model that used the MH algorithm, that is, the acceptance decision using Bern(zn∣rnMH), was closer to the behavior of participants than that using several heuristic comparative models. We performed a test using the assessment of acceptance or rejection obtained from the results of the communication experiment, and the inferred acceptance probability was denoted as rnMH. We created a dataset consisting of distances between the behaviors of participants and the samples generated from the probabilities calculated by the five comparison models. These models were used to evaluate the acceptance and rejection. Subsequently, U-tests were conducted for each model.

Table 3 lists the comparative models used in this study. Constant accepts with a probability b̄ calculated from the actual acceptance rate of the subject from the experimental results, which corresponds to the null hypothesis of Hypothesis testing 1. MH accepts with the inferred MH-based acceptance probability rnMH from the experimental results. Numerator accepts with the acceptance probability being the numerator part of the rnMH score, which represents the likelihood of the sign of the opponent using its own parameter. Subtraction calculates the difference between the numerator part of the rnMH score representing the likelihood of the sign of the opponent using the parameter of the listener and the denominator part representing the likelihood of its own sign instead of the ratio in rnMH score. Subsequently, it was transformed into a range of 0.0–1.0. Binary accepts with a probability of 0.1 if the inferred acceptance probability rnMH is less than or equal to 0.5 and 0.9 if it exceeds 0.5.

To test the statistical significance of models m and m′ that make decisions regarding acceptance and rejection, hypothesis tests were performed as null and alternative hypotheses, respectively, as follows:

Here, Prec_m is the rate at which the model m could predict the acceptance or rejection decisions of participants, that is, precision.

We sampled 100 data points for the pseudo-experimental results of each comparison model using computer simulations. The pseudo-experimental results for each comparison model were sampled from the Bernoulli distribution with the parameter of acceptance probability r(j,i)m for subject j of model m in the ith communication trial and labeled 1 for acceptance and 0 for rejection. The p values were calculated using a U-test, and the significance level was set at 0.001.

Precision was calculated as follows: First, we store the acceptance/rejection evaluation of the j-th participant at the ith trial in the experiment in z(j,i)h, where j = 1, ⋯, 20. Second, we store the acceptance/rejection evaluation results of model m in the i-th trial of the pseudo-experiment for subject j in z(j,i)m, where i = 1, ⋯, 45 (i = 1, ⋯, 90 for both datasets). Third, we calculate the precision of model m in predicting the behavior of the j-th participant.

Precision Prec_m is calculated by counting the number of matches between the decisions of the participants and the model. One-sided tests were conducted for all model combinations.

Figure 7 illustrates an example of the actual acceptance/rejection behavior of a participant and the inferred acceptance probabilities r^MH. This suggests some coherence between rnMH and the behavior of participants. This association was evaluated quantitatively and statistically.

Figure 8 shows a histogram of the number of accepted stimuli for each acceptance rate (left) and the actual acceptance rate for each acceptance rate with a graph of y = ar+b using the estimated weights a and bias b (right) for all the participants, where a = 0.5105 and b = 0.4842. When the inferred acceptance rate was high, the actual acceptance rate by humans was also high. However, the actual probability of acceptance was higher than r^MH when r^MH was low. It was rare for the inferred acceptance rate, rnMH to assume intermediate values between 0.2 and 0.8. This bias is inherent in the nature of the MHNG and does not reflect the characteristics of human participants. In the MH algorithm, acceptance probability is determined by the ratio of one probability to another, which often results in values close to 0 or 1. For further clarification, Figure 9 shows the acceptance rate as observed in computer simulations for comparison.

Here, we discuss the results of the hypotheses tests. First, we examine the results of Test 1. The estimated parameters for Datasets 1 and 2 are shown in Table 4. The hypothesis tests for each and both datasets for all subjects were rejected at a significance level of 0.001. Therefore, the null hypothesis was rejected, indicating that the model using acceptance probability computed by the MH algorithm is a significantly better predictor of human behavior than the model using a constant probability of acceptance.

For more detail, we examined each participant's behavior. The p-values P_a(a) for each subject, as presented in Table 4, indicate that the null hypothesis was rejected at the 0.05 significance level for most subjects. Specifically, exceptions were found for two participants, namely 10 and 18, in dataset 1. In dataset 2, all results were rejected at the 0.05 significance level except for the results for participants 5, 6, 9, 12, 14, and 16. Overall, the null hypothesis was rejected in most cases.

Here, we examine the results of Test 2. Table 5 shows the p-values obtained from the U-tests conducted for each combination of models. The row for MH (i.e., m = 2) in Table 5 shows that the null hypothesis was rejected for all the models. The model using the MH algorithm was the closest to the behaviors of participants among the models compared in this study. We also individually performed tests on data from each participant. Table 6 presents the results. For each participant, MH outperformed the other models in predicting behavior in all cases, except for six participants in Constant and one in Subtraction. For the six participants, MH did not significantly outperform Constant, and for one participant, MH did not significantly outperform Subtraction. We tested the data for each participant separately and for each dataset. Tables 7, 8 list the results. Looking at the MH (i.e., m = 2) row in Table 7, MH outperformed the other models in all cases except seven for Constant and one for Numerator. Looking at the MH (i.e., m = 2) row in Table 8, MH outperformed the other models in all cases, except five for Constant. Based on these results, we argue that the acceptance probability derived from the MH algorithm is, to some extent, consistent with the acceptance/rejection judgment probabilities exhibited by humans.

The experimental results suggested that human behavior in the JA-NG follows the MH algorithm. Consequently, this result suggests that symbol emergence in the JA-NG among people may be attained by performing decentralized Bayesian inference, i.e., collective predictive coding (Taniguchi, 2023).