# Opinion Dynamics with Bayesian Learning

Authors: Aili Fang; Kehua Yuan; Jinhua Geng; Xinjiang Wei
Journal: Complexity
 (2020)

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## Abstract

Bayesian learning is a rational and effective strategy in the opinion dynamic process. In this paper, we theoretically prove that individual Bayesian learning can realize asymptotic learning and we test it by simulations on the Zachary network. Then, we propose a Bayesian social learning model with signal update strategy and apply the model on the Zachary network to observe opinion dynamics. Finally, we contrast the two learning strategies and find that Bayesian social learning can lead to asymptotic learning more faster than individual Bayesian learning.


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## Body

## 1. Introduction

We all have our own opinions on various topics of social issues. The opinions are formed by an evolutionary process in social context [1–3]. Opinion dynamics is the study of the opinion fusion process through interactions among a group of agents [4]. Some interesting models have been proposed, such as the DeGroot model [5], voter model [6, 7], bounded confidence model [8–10], and many other models [11–14]. Among these models, a key element is the opinion update strategy in the dynamic process. All the opinion update strategies can be classified into two categories by using Bayesian update rule or not: one is Bayesian update strategy and the other is non-Bayesian strategy. Non-Bayesian learning refers to individuals updating their opinions by non-Bayesian strategy, such as linear combination of opinions of the neighbors [5, 15, 16] and various game theories in social network [17, 18]. Most of these models try to explore how the society could achieve group consensus. But the consensus opinion might not necessarily be the truth; in other words, the true state might not be realized. While Bayesian learning assumes that individual updates their opinions according to Bayesian rule, individuals can learn the truth in the long run according to the prior information [19–21]. Therefore, Bayesian learning can integrate the prior information and lead to realize the truth more rationally and effectively than non-Bayesian learning.Though Bayesian learning can achieve the truth, there is no systematical proof in the existing literature [22, 23]. The proof of individual Bayesian learning lays a theoretical foundation for the learning process, so in this paper, we will deduce the truthfulness of individual Bayesian learning by theoretical derivation. Moreover, considering people’s crowd psychology, we put forward a signal update strategy which means that people adjust their observations to meet with the most people. Combined with this signal update strategy, we propose a Bayesian social learning model to study the opinion dynamics under social environment. Furthermore, we conduct simulations on Zachary network to observe the learning results.The rest of the paper is organized as follows. In Section2, we give a theoretical proof of individual Bayesian learning. In Section 3, we propose a Bayesian social learning model with signal update strategy and test it by simulations on Zachary network. In Section 4, we draw conclusions.


## 2. Individual Bayesian Learning

### 2.1. Individual Bayesian Learning Model

Let state spaceΘ=θ1,θ2,…,θm and the underlying true state be θ∗θ∗∈Θ. Individual i’s opinion on state θk at time t is denoted by probability distribution μi,tθkk=1,2,…,m, where m is a finite integer. At each time period, the signal set St=St1,St2,…,Stn∈S1×⋯×Sn is generated by the likelihood function PStθk conditional on state θk, where Sti∈Si denotes the signal privately observed by individual i at time t and Si denotes the individual i’s signal space. The ith marginal of Pθk is denoted by Piθk which is known as individual i’s signal structure conditional on state θk. We assume that each individual’s private signal structure is commonly known. At each time, individual i receives his private signal Sti and updates his prior opinion to the posterior opinion by Bayesian law. Bayesian statistics combining prior information can make the inference results more accurate and effective [24]. As time goes by, the individual’s opinion will show some amazing dynamics during the evolution process. Next, we will give the denotations of asymptotic learning which is an important definition in opinion dynamics.Definition 1.
Asymptotic learning: asymptotic learning refers to that individuali who receives the signals generated by true state θ∗∈Θ and achieves asymptotic learning on a path Stit=1∞ if along that path μi,tθ∗⟶1 with probability one as t⟶∞.


### 2.2. Theorem and Proof

Theorem 1.
Assume that(1)
Individuali has positive prior belief on true state θ∗, i.e., μi,0θ∗>0.(2)
At each timet, ∀Sti∈St,PStiθ>0 holds for all i and θ.(3)
There is no observationally equivalent state toθ∗ for individual i, i.e., θ∈ΘPisθ=Pisθ∗,∀s∈Si=∅.
At each time stept, individual i has his prior opinion μi,tθ on state θ; after receiving signal St+1i which is generated by true state θ∗∈Θ, his opinion μi,t+1θ at the next time step t+1 will be updated to his posterior probability by Bayesian law:(1)μi,t+1θ=μi,t+1θS1i,S2i,…,Sti,St+1i=μi,tθPS1i,S2i,…,Sti,St+1iθPS1i,S2i,…,Sti,St+1iθ∈Θ.Then, μi,tθ∗⟶1 will hold with probability one as t⟶∞ and we call individual i realizes asymptotic learning.Proof.
Let state spaceΘ=θ1,θ2,…,θm and the true state θ∗=θ1. Suppose that the signal sequences Stit=1,2,… are independent and identically distributed given θk, then we have(2)PS1i,S2i,…,Stiθk=∏r=1tPSriθk.
At timet+1, individual i updates his opinion and obtains his posterior opinion by Bayesian rule (1) after receiving signal St+1i.(3)μi,t+1θk=μi,t+1θkS1i,S2i,…,Sti,St+1i=μi,tθkPS1i,S2i,…,Sti,St+1iθkPS1i,S2i,…,Sti,St+1i=μi,tθk∏r=1t+1PSriθkΣθϵΘμi,tθ∏r=1t+1PSriθ.
Next, we compare the probability of two state of the partition, sayθ1 and θ2 in the light of St+1i for individual i at time t+1,(4)μi,t+1θ1μi,t+1θ2=μi,tθ1μi,tθ2∏r=1t+1PSriθ1PSriθ2=μi,tθ1μi,tθ2∏r=1t+1δSri=μi,tθ1μi,tθ2RSt+1i,where δSri is the likelihood ration of θ1 to θ2 given Sri and RSt+1i is the likelihood ration of θ1 to θ2 given S1i,S2i,…,St+1i.
SupposePδSri<∞θ1=ϕ, two cases are to be distinguished.
In the first case, supposeϕ<1, then PRSti=∞θ1=1−ϕt which obviously approaches 1 with increasing t. Another forms of expression can be(5)limt⟶∞PRSti≥ρθ1=1for0≤ρ<∞.
Equation (5) shows that the probability that PRSti given θ1 is greater than any preassigned number is almost 1.
The second case isϕ=1. Since much is known about sums of identically distributed independent random variables, it is natural to investigate(6)logRSt+1i=∑r=1t+1logδSri,thereby replacing a product by a sum. It is easily seen from the definition of δSri that δSri>0θ1=1, so for the case now at hand, the function logδSri is independent real bounded random variable.
LettingI=ElogδSriθ1, the weak law of large numbers implies that, for any ε>0,(7)limt⟶∞P1t+1∑i=1t+1logRSt+1i−ElogRSt+1i≤εθ1=1,and it can be transformed into(8)limt⟶∞PlogRSt+1i≥t+1I−εθ1=1.Equivalently,(9)limt⟶∞PRSt+1i≥et+1I−εθ1=1.
According to expectation inequality [25], consider that(10)I=ElogδSriθ1≥logE−1δ−1Sriθ1=−logEPSriθ2PSriθ1θ1=−log∑Sri∈St+1iPSriθ2PSriθ1⋅PSriθ1=−log∑Sri∈St+1iPSriθ2=−log1=0,the equality can hold in (10) if and only if δ−1Sri is constant with probability 1, given θ1. Since the expected value of δ−1Sri is equal to 1, the equality will hold if and only if Pδ−1Sriθ1=1. This means state θ2 is observationally equivalent state to θ1, which is contradictory with condition (3) of the theorem. So, I>0. Then, according to (9), we can also infer that (5) holds.
Therefore, under the assumption conditions of the theorem, we can demonstratePRSti=∞θ1=1 holds with probability one; consequently, according to (10), the ratio of individual i′s posterior probability of the real state θ∗=θ1 to the other state θ2 tends to infinity, i.e.,(11)μi,t+1θ∗μi,t+1θ2⟶∞.
By the same method, it can be proved that the likelihood ratio functions of real statesθ∗ to the other states θk,k=3,4,…,m also tend to infinity, i.e.,(12)μi,t+1θ∗μi,t+1θk⟶∞.
Sinceμi,t+1θ∗+∑i=2mμi,t+1θk=1 and 0≤μi,t+1θ≤1∀θ∈Θ, we will have μi,t+1θ∗⟶1 and μi,t+1θk⟶0∀θk≠θ∗ when t⟶∞.
In summary, the individual can become highly convinced of the truth by the Bayesian law and achieve asymptotic learning after abundant observations.


### 2.3. Simulation

We have theoretically proved that individuals can use Bayesian law to update their opinions and achieve asymptotic learning. Next, we will verify the truth of the above individual Bayesian learning model by simulations on individuals in Zachary network.


#### 2.3.1. Initial Values Setting

Before conducting the numerical simulations, some assumptions are given as follows:(1)
Different signals are independent of each other(2)
The connection between individuals is indirect(3)
In the initial state, all individuals have the same opinion in different states(4)
Each individual observes only one signal at a time(5)
Different individuals have the same signal structureThis simulation experiment mainly aims at social learning on two states. Now, we suppose thatΘ=θ1,θ2 and the true state θ∗=θ1, and the signal space S=S1,S2,…,S5. The individual composition and their relationship are shown in Figure 1, which is famously known as Zachary network [26].Figure 1
Zachary network.At timet=0, individual i’s opinion on θ1 and θ2 is μi,0θ1 and μi,0θ2, respectively. Under assumption (3), the individuals’ initial opinions are set to be(13)μ1,0θ1,μ2,0θ1,…,μ34,0θ1=0.5,0.5,…,0.5,μ1,0θ2,μ2,0θ2,…,μ34,0θ2=0.5,0.5,…,0.5.Under assumption (5), the signal structures are given for all i as follows and they will remain unchanged during the learning process:(14)PiS1θ1,PiS2θ1,…,PiS5θ1=0.35,0.20,0.15,0.05,0.25,PiS1θ2,PiS2θ2,…,PiS5θ2=0.30,0.15,0.10,0.25,0.20.In the individual Bayesian learning model, it is assumed that individual opinion evolution is affected by their prior knowledge and signal characteristics. When the individuali’s opinion on the underlying true state μi,tθ∗ is larger than 0.9999, we consider that he reaches asymptotic learning. If each individual realizes asymptotic learning, the whole society will form a social consensus and find the truth.


#### 2.3.2. Simulation Results

Under the above initial conditions, we conduct simulations on Zachary network, and results in Figure2 show that μi,tθ1⟶1 and μi,tθ2⟶0 when t⟶∞ for all i, which means all the individuals can achieve asymptotic learning and realize the true state.Figure 2
Individual Bayesian learning results.
(a)
(b)


## 2.1. Individual Bayesian Learning Model

Let state spaceΘ=θ1,θ2,…,θm and the underlying true state be θ∗θ∗∈Θ. Individual i’s opinion on state θk at time t is denoted by probability distribution μi,tθkk=1,2,…,m, where m is a finite integer. At each time period, the signal set St=St1,St2,…,Stn∈S1×⋯×Sn is generated by the likelihood function PStθk conditional on state θk, where Sti∈Si denotes the signal privately observed by individual i at time t and Si denotes the individual i’s signal space. The ith marginal of Pθk is denoted by Piθk which is known as individual i’s signal structure conditional on state θk. We assume that each individual’s private signal structure is commonly known. At each time, individual i receives his private signal Sti and updates his prior opinion to the posterior opinion by Bayesian law. Bayesian statistics combining prior information can make the inference results more accurate and effective [24]. As time goes by, the individual’s opinion will show some amazing dynamics during the evolution process. Next, we will give the denotations of asymptotic learning which is an important definition in opinion dynamics.Definition 1.
Asymptotic learning: asymptotic learning refers to that individuali who receives the signals generated by true state θ∗∈Θ and achieves asymptotic learning on a path Stit=1∞ if along that path μi,tθ∗⟶1 with probability one as t⟶∞.


## 2.2. Theorem and Proof

Theorem 1.
Assume that(1)
Individuali has positive prior belief on true state θ∗, i.e., μi,0θ∗>0.(2)
At each timet, ∀Sti∈St,PStiθ>0 holds for all i and θ.(3)
There is no observationally equivalent state toθ∗ for individual i, i.e., θ∈ΘPisθ=Pisθ∗,∀s∈Si=∅.
At each time stept, individual i has his prior opinion μi,tθ on state θ; after receiving signal St+1i which is generated by true state θ∗∈Θ, his opinion μi,t+1θ at the next time step t+1 will be updated to his posterior probability by Bayesian law:(1)μi,t+1θ=μi,t+1θS1i,S2i,…,Sti,St+1i=μi,tθPS1i,S2i,…,Sti,St+1iθPS1i,S2i,…,Sti,St+1iθ∈Θ.Then, μi,tθ∗⟶1 will hold with probability one as t⟶∞ and we call individual i realizes asymptotic learning.Proof.
Let state spaceΘ=θ1,θ2,…,θm and the true state θ∗=θ1. Suppose that the signal sequences Stit=1,2,… are independent and identically distributed given θk, then we have(2)PS1i,S2i,…,Stiθk=∏r=1tPSriθk.
At timet+1, individual i updates his opinion and obtains his posterior opinion by Bayesian rule (1) after receiving signal St+1i.(3)μi,t+1θk=μi,t+1θkS1i,S2i,…,Sti,St+1i=μi,tθkPS1i,S2i,…,Sti,St+1iθkPS1i,S2i,…,Sti,St+1i=μi,tθk∏r=1t+1PSriθkΣθϵΘμi,tθ∏r=1t+1PSriθ.
Next, we compare the probability of two state of the partition, sayθ1 and θ2 in the light of St+1i for individual i at time t+1,(4)μi,t+1θ1μi,t+1θ2=μi,tθ1μi,tθ2∏r=1t+1PSriθ1PSriθ2=μi,tθ1μi,tθ2∏r=1t+1δSri=μi,tθ1μi,tθ2RSt+1i,where δSri is the likelihood ration of θ1 to θ2 given Sri and RSt+1i is the likelihood ration of θ1 to θ2 given S1i,S2i,…,St+1i.
SupposePδSri<∞θ1=ϕ, two cases are to be distinguished.
In the first case, supposeϕ<1, then PRSti=∞θ1=1−ϕt which obviously approaches 1 with increasing t. Another forms of expression can be(5)limt⟶∞PRSti≥ρθ1=1for0≤ρ<∞.
Equation (5) shows that the probability that PRSti given θ1 is greater than any preassigned number is almost 1.
The second case isϕ=1. Since much is known about sums of identically distributed independent random variables, it is natural to investigate(6)logRSt+1i=∑r=1t+1logδSri,thereby replacing a product by a sum. It is easily seen from the definition of δSri that δSri>0θ1=1, so for the case now at hand, the function logδSri is independent real bounded random variable.
LettingI=ElogδSriθ1, the weak law of large numbers implies that, for any ε>0,(7)limt⟶∞P1t+1∑i=1t+1logRSt+1i−ElogRSt+1i≤εθ1=1,and it can be transformed into(8)limt⟶∞PlogRSt+1i≥t+1I−εθ1=1.Equivalently,(9)limt⟶∞PRSt+1i≥et+1I−εθ1=1.
According to expectation inequality [25], consider that(10)I=ElogδSriθ1≥logE−1δ−1Sriθ1=−logEPSriθ2PSriθ1θ1=−log∑Sri∈St+1iPSriθ2PSriθ1⋅PSriθ1=−log∑Sri∈St+1iPSriθ2=−log1=0,the equality can hold in (10) if and only if δ−1Sri is constant with probability 1, given θ1. Since the expected value of δ−1Sri is equal to 1, the equality will hold if and only if Pδ−1Sriθ1=1. This means state θ2 is observationally equivalent state to θ1, which is contradictory with condition (3) of the theorem. So, I>0. Then, according to (9), we can also infer that (5) holds.
Therefore, under the assumption conditions of the theorem, we can demonstratePRSti=∞θ1=1 holds with probability one; consequently, according to (10), the ratio of individual i′s posterior probability of the real state θ∗=θ1 to the other state θ2 tends to infinity, i.e.,(11)μi,t+1θ∗μi,t+1θ2⟶∞.
By the same method, it can be proved that the likelihood ratio functions of real statesθ∗ to the other states θk,k=3,4,…,m also tend to infinity, i.e.,(12)μi,t+1θ∗μi,t+1θk⟶∞.
Sinceμi,t+1θ∗+∑i=2mμi,t+1θk=1 and 0≤μi,t+1θ≤1∀θ∈Θ, we will have μi,t+1θ∗⟶1 and μi,t+1θk⟶0∀θk≠θ∗ when t⟶∞.
In summary, the individual can become highly convinced of the truth by the Bayesian law and achieve asymptotic learning after abundant observations.


## 2.3. Simulation

We have theoretically proved that individuals can use Bayesian law to update their opinions and achieve asymptotic learning. Next, we will verify the truth of the above individual Bayesian learning model by simulations on individuals in Zachary network.


### 2.3.1. Initial Values Setting

Before conducting the numerical simulations, some assumptions are given as follows:(1)
Different signals are independent of each other(2)
The connection between individuals is indirect(3)
In the initial state, all individuals have the same opinion in different states(4)
Each individual observes only one signal at a time(5)
Different individuals have the same signal structureThis simulation experiment mainly aims at social learning on two states. Now, we suppose thatΘ=θ1,θ2 and the true state θ∗=θ1, and the signal space S=S1,S2,…,S5. The individual composition and their relationship are shown in Figure 1, which is famously known as Zachary network [26].Figure 1
Zachary network.At timet=0, individual i’s opinion on θ1 and θ2 is μi,0θ1 and μi,0θ2, respectively. Under assumption (3), the individuals’ initial opinions are set to be(13)μ1,0θ1,μ2,0θ1,…,μ34,0θ1=0.5,0.5,…,0.5,μ1,0θ2,μ2,0θ2,…,μ34,0θ2=0.5,0.5,…,0.5.Under assumption (5), the signal structures are given for all i as follows and they will remain unchanged during the learning process:(14)PiS1θ1,PiS2θ1,…,PiS5θ1=0.35,0.20,0.15,0.05,0.25,PiS1θ2,PiS2θ2,…,PiS5θ2=0.30,0.15,0.10,0.25,0.20.In the individual Bayesian learning model, it is assumed that individual opinion evolution is affected by their prior knowledge and signal characteristics. When the individuali’s opinion on the underlying true state μi,tθ∗ is larger than 0.9999, we consider that he reaches asymptotic learning. If each individual realizes asymptotic learning, the whole society will form a social consensus and find the truth.


### 2.3.2. Simulation Results

Under the above initial conditions, we conduct simulations on Zachary network, and results in Figure2 show that μi,tθ1⟶1 and μi,tθ2⟶0 when t⟶∞ for all i, which means all the individuals can achieve asymptotic learning and realize the true state.Figure 2
Individual Bayesian learning results.
(a)
(b)


## 2.3.1. Initial Values Setting

Before conducting the numerical simulations, some assumptions are given as follows:(1)
Different signals are independent of each other(2)
The connection between individuals is indirect(3)
In the initial state, all individuals have the same opinion in different states(4)
Each individual observes only one signal at a time(5)
Different individuals have the same signal structureThis simulation experiment mainly aims at social learning on two states. Now, we suppose thatΘ=θ1,θ2 and the true state θ∗=θ1, and the signal space S=S1,S2,…,S5. The individual composition and their relationship are shown in Figure 1, which is famously known as Zachary network [26].Figure 1
Zachary network.At timet=0, individual i’s opinion on θ1 and θ2 is μi,0θ1 and μi,0θ2, respectively. Under assumption (3), the individuals’ initial opinions are set to be(13)μ1,0θ1,μ2,0θ1,…,μ34,0θ1=0.5,0.5,…,0.5,μ1,0θ2,μ2,0θ2,…,μ34,0θ2=0.5,0.5,…,0.5.Under assumption (5), the signal structures are given for all i as follows and they will remain unchanged during the learning process:(14)PiS1θ1,PiS2θ1,…,PiS5θ1=0.35,0.20,0.15,0.05,0.25,PiS1θ2,PiS2θ2,…,PiS5θ2=0.30,0.15,0.10,0.25,0.20.In the individual Bayesian learning model, it is assumed that individual opinion evolution is affected by their prior knowledge and signal characteristics. When the individuali’s opinion on the underlying true state μi,tθ∗ is larger than 0.9999, we consider that he reaches asymptotic learning. If each individual realizes asymptotic learning, the whole society will form a social consensus and find the truth.


## 2.3.2. Simulation Results

Under the above initial conditions, we conduct simulations on Zachary network, and results in Figure2 show that μi,tθ1⟶1 and μi,tθ2⟶0 when t⟶∞ for all i, which means all the individuals can achieve asymptotic learning and realize the true state.Figure 2
Individual Bayesian learning results.
(a)
(b)


## 3. Bayesian Social Learning Model

### 3.1. Bayesian Social Learning

In the individual Bayesian learning process, individuals update their opinions by Bayesian law to achieve asymptotic learning. But the whole society is a complex social network, and individuals are connected and have influences on each other. By communication with his neighbors, the individual can have knowledge of his neighbors’ received signals and their opinions. So, the individual will continuously adjust his opinion according to the signals received not only by himself but also by his neighbors. Therefore, we consider opinion dynamics with Bayesian law in a social network background, which is called Bayesian social learning.Here, the social network is abstracted into a graphG=V,E composed of individuals and their interactions. Let V represent the set of all individuals and E represent the set of relationship between every two individuals. At time t+1, each individual i receives a signal St+1i whose distribution follows his signal structure Pi⋅θ∗. Because the individuals have influence on each other and people have herd mentality, then the individual i will adjust his signal to the signal S^t+1 which is received by the most people. Therefore, individuals update their signals by the following rule:(15)St+1i=St+1i,ifSt+1i=S^t+1,S^t+1,ifSt+1i≠S^t+1.After updating signals, individuali will update his own opinion about state θ by Bayesian law as follows:(16)μi,t+1θ=μi,t+1θS1i,S2i,…,Sti,St+1i=μi,tθPS1i,S2i,…,Sti,St+1iθPS1i,S2i,…,Sti,St+1iθ∈Θ.Next, we will explore the social learning results in the opinion dynamic process according to the signal update rule (15) and opinion update rule (16) by simulations.


### 3.2. Simulation Results

We also take Zachary network as an example society. The state spaceΘ=θ1,θ1, and signal structure and initial opinions μi,0θ are set to be the same as the previous simulations. The Bayesian social learning results are shown in Figure 3.Figure 3
Bayesian social learning results.
(a)
(b)It is obviously seen that individuals can realize asymptotic learning by Bayesian social learning with signal update strategy. The strong connectivity of the network helps the group in the network to reach social consensus quickly with little fluctuation. Furthermore, we compare the average consensus time of Bayesian social learning and individual Bayesian learning as shown in Table1.Table 1
Comparison of average consensus time.
Model
Consensus time
Individual Bayesian learning model
98
Bayesian social learning model
61We can see that Bayesian social learning with the signal update strategy model has a significantly faster learning speed than the individual Bayesian learning model, and it has less fluctuation. So, we can speculate that the interactions among individuals in the society might accelerate information congregation and achieve asymptotic learning faster.


## 3.1. Bayesian Social Learning

In the individual Bayesian learning process, individuals update their opinions by Bayesian law to achieve asymptotic learning. But the whole society is a complex social network, and individuals are connected and have influences on each other. By communication with his neighbors, the individual can have knowledge of his neighbors’ received signals and their opinions. So, the individual will continuously adjust his opinion according to the signals received not only by himself but also by his neighbors. Therefore, we consider opinion dynamics with Bayesian law in a social network background, which is called Bayesian social learning.Here, the social network is abstracted into a graphG=V,E composed of individuals and their interactions. Let V represent the set of all individuals and E represent the set of relationship between every two individuals. At time t+1, each individual i receives a signal St+1i whose distribution follows his signal structure Pi⋅θ∗. Because the individuals have influence on each other and people have herd mentality, then the individual i will adjust his signal to the signal S^t+1 which is received by the most people. Therefore, individuals update their signals by the following rule:(15)St+1i=St+1i,ifSt+1i=S^t+1,S^t+1,ifSt+1i≠S^t+1.After updating signals, individuali will update his own opinion about state θ by Bayesian law as follows:(16)μi,t+1θ=μi,t+1θS1i,S2i,…,Sti,St+1i=μi,tθPS1i,S2i,…,Sti,St+1iθPS1i,S2i,…,Sti,St+1iθ∈Θ.Next, we will explore the social learning results in the opinion dynamic process according to the signal update rule (15) and opinion update rule (16) by simulations.


## 3.2. Simulation Results

We also take Zachary network as an example society. The state spaceΘ=θ1,θ1, and signal structure and initial opinions μi,0θ are set to be the same as the previous simulations. The Bayesian social learning results are shown in Figure 3.Figure 3
Bayesian social learning results.
(a)
(b)It is obviously seen that individuals can realize asymptotic learning by Bayesian social learning with signal update strategy. The strong connectivity of the network helps the group in the network to reach social consensus quickly with little fluctuation. Furthermore, we compare the average consensus time of Bayesian social learning and individual Bayesian learning as shown in Table1.Table 1
Comparison of average consensus time.
Model
Consensus time
Individual Bayesian learning model
98
Bayesian social learning model
61We can see that Bayesian social learning with the signal update strategy model has a significantly faster learning speed than the individual Bayesian learning model, and it has less fluctuation. So, we can speculate that the interactions among individuals in the society might accelerate information congregation and achieve asymptotic learning faster.


## 4. Conclusion

In this paper, we research on two Bayesian learning models. The first model is the individual Bayesian learning model in which we deduce the truthfulness of individual Bayesian learning by theoretical derivation. Moreover, the numerical simulations also show that individuals who update their opinions by Bayesian law can realize asymptotic learning.Furthermore, we propose the Bayesian social learning model with signal update strategy and test it by simulations on Zachary network. The results show that individuals adopting the signal update strategy proposed in this paper can realize asymptotic learning similarly.We compare the results of two models and find that Bayesian social learning model can achieve asymptotic learning more faster under the same conditions. In the future study, we will explore the theoretical supports for the Bayesian social learning model.



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