# 社会学习数学建模

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**Mathematical models of social learning** aim to model opinion dynamics in social networks. Consider a social network in which people (agents) hold a belief or opinion about the state of something in the world, such as the quality of a particular product, the effectiveness of a public policy, or the reliability of a news agency. In all these settings, people learn about the state of the world via observation or communication with others. Models of social learning try to formalize these interactions to describe how agents process the information received from their friends in the social network. Some of the main questions asked in the literature include:^{[1]}

Mathematical models of social learning aim to model opinion dynamics in social networks. Consider a social network in which people (agents) hold a belief or opinion about the state of something in the world, such as the quality of a particular product, the effectiveness of a public policy, or the reliability of a news agency. In all these settings, people learn about the state of the world via observation or communication with others. Models of social learning try to formalize these interactions to describe how agents process the information received from their friends in the social network. Some of the main questions asked in the literature include:

社会学习的数学模型旨在建立社会网络中的意见动态模型。考虑一个社交网络，在这个网络中，人们(代理人)对世界上某种事物的状态持有一种信念或观点，例如某种特定产品的质量、公共政策的有效性或新闻机构的可靠性。在所有这些环境中，人们通过观察或与他人交流来了解世界的状态。社会学习模型试图将这些互动形式化，以描述代理人如何处理从社会网络中的朋友那里接收到的信息。文献中提出的一些主要问题包括:

- whether agents reach a consensus;
- whether social learning effectively aggregates scattered information, or put differently, whether the consensus belief matches the true state of the world or not;
- how effective media sources, politicians, and prominent agents can be in belief formation of the entire network. In other words, how much room is there for belief manipulation and misinformation?

- whether agents reach a consensus;
- whether social learning effectively aggregates scattered information, or put differently, whether the consensus belief matches the true state of the world or not;
- how effective media sources, politicians, and prominent agents can be in belief formation of the entire network. In other words, how much room is there for belief manipulation and misinformation?

- 是否代理人达成共识 # 是否社会学习有效地聚合零散的信息，或换句话说，共识信念是否符合世界的真实状态 # 如何有效的媒体来源，政治家，和杰出的代理人可以在信念形成的整个网络。换句话说，信仰操纵和错误信息的空间有多大？

## Bayesian learning

Bayesian learning is a model which assumes that agents update their beliefs using Bayes' rule. Indeed, each agent's belief about different states of the world can be seen as a probability distribution over a set of opinions, and Bayesian updating assumes that this distribution is updated in a statistically optimal manner using Bayes' rule. Moreover, Bayesian models typically make certain demanding assumptions about agents, e.g., that they have a reliable model of the world and that the social learning rule of each agent is common knowledge among all members of the community.

Bayesian learning is a model which assumes that agents update their beliefs using Bayes' rule. Indeed, each agent's belief about different states of the world can be seen as a probability distribution over a set of opinions, and Bayesian updating assumes that this distribution is updated in a statistically optimal manner using Bayes' rule. Moreover, Bayesian models typically make certain demanding assumptions about agents, e.g., that they have a reliable model of the world and that the social learning rule of each agent is common knowledge among all members of the community.

贝叶斯学习是一种模型，它假定代理人使用贝叶斯规则更新他们的信念。事实上，每个代理人对世界不同状态的信念可以被看作是一系列观点的概率分布，而贝叶斯更新假设这种分布是使用贝叶斯规则以统计学上最优的方式更新的。此外，贝叶斯模型通常对代理人做出某些苛刻的假设，例如，他们有一个可靠的世界模型，每个代理人的社会学习规则是社区所有成员的共同知识。

More rigorously, let the underlying state be θ. This parameter could correspond to an opinion among people about a certain social, economic, or political issue. At first, each individual has a prior probability of θ which can be shown by P(θ). This prior could be a result of the agents' personal observations of the world. Then each person updates their belief by receiving some signal *s*. According to the Bayesian approach, the updating procedure will follow this rule:

More rigorously, let the underlying state be θ. This parameter could correspond to an opinion among people about a certain social, economic, or political issue. At first, each individual has a prior probability of θ which can be shown by P(θ). This prior could be a result of the agents' personal observations of the world. Then each person updates their belief by receiving some signal s. According to the Bayesian approach, the updating procedure will follow this rule:

更严格地说，让底层状态为 θ。这个参数可能与人们对某一社会、经济或政治问题的看法相对应。首先，每个个体都有一个可以用 p (θ)表示的 θ 先验概率。这种先验可能是代理人对世界的个人观察的结果。然后每个人通过接收一些信号来更新他们的信念。根据贝叶斯方法，更新程序将遵循以下规则:

[math]\displaystyle{ P(\theta|s) = \frac{P(s|\theta)}{P(s)} \cdot P(\theta) }[/math]

P(\theta|s) = \frac{P(s|\theta)}{P(s)} \cdot P(\theta)

P (theta | s) = frac { p (s | theta)}{ p (s)} cdot p (theta)

where the term [math]\displaystyle{ \textstyle P(s|\theta) }[/math] is the conditional probability over signal space given the true state of the world.^{[1]}

where the term \textstyle P(s|\theta) is the conditional probability over signal space given the true state of the world.

其中短信样式 p (s | theta)是给定世界真实状态的信号空间上的条件概率。

## Non-Bayesian learning

Bayesian learning is often considered the benchmark model for social learning, in which individuals use Bayes' rule to incorporate new pieces of information to their belief. However, it has been shown that such a Bayesian "update" is fairly sophisticated and imposes an unreasonable cognitive load on agents which might not be realistic for human beings.^{[2]}

Bayesian learning is often considered the benchmark model for social learning, in which individuals use Bayes' rule to incorporate new pieces of information to their belief. However, it has been shown that such a Bayesian "update" is fairly sophisticated and imposes an unreasonable cognitive load on agents which might not be realistic for human beings.

= = 非贝叶斯学习 = = 贝叶斯学习通常被认为是社会学习的基准模型，在这个模型中，个体使用贝叶斯规则将新的信息融入到他们的信念中。然而，已经证明这样的贝叶斯“更新”是相当复杂的，并强加了不合理的认知负荷的代理人可能是不现实的人类。

Therefore, scientists have studied simpler non-Bayesian models, most notably the DeGroot model, introduced by DeGroot in 1974, which is one of the very first models for describing how humans interact with each other in a social network. In this setting, there is a true state of the world, and each agent receives a noisy independent signal from this true value and communicates with other agents repeatedly. According to the DeGroot model, each agent takes a weighted average of their neighbors' opinions at each step to update their own belief.

Therefore, scientists have studied simpler non-Bayesian models, most notably the DeGroot model, introduced by DeGroot in 1974, which is one of the very first models for describing how humans interact with each other in a social network. In this setting, there is a true state of the world, and each agent receives a noisy independent signal from this true value and communicates with other agents repeatedly. According to the DeGroot model, each agent takes a weighted average of their neighbors' opinions at each step to update their own belief.

因此，科学家们研究了更简单的非贝叶斯模型，其中最引人注目的是德格鲁特模型(DeGroot model) ，该模型由德格鲁特于1974年提出，是描述人类在社交网络中如何相互交流的最早模型之一。在这种情况下，存在一个真实的世界状态，每个代理从这个真实值接收一个噪声独立的信号，并与其他代理重复通信。根据 DeGroot 模型，每个代理人在每一步都会收集他们邻居的意见，并更新他们自己的加权平均数。

The statistician George E. P. Box once said, "All models are wrong; however, some of them are useful." Along the same lines, the DeGroot model is a fairly simple model but it can provide us with useful insights about the learning process in social networks. Indeed, the simplicity of this model makes it tractable for theoretical studies. Specifically, we can analyze different network structure to see for which structures these naive agents can successfully aggregate decentralized information. Since the DeGroot model can be considered a Markov chain, provided that a network is strongly connected (so there is a direct path from any agent to any other) and satisfies a weak aperiodicity condition, beliefs will converge to a consensus. When consensus is reached, the belief of each agent is a weighted average of agents' initial beliefs. These weights provide a measure of social influence.

The statistician George E. P. Box once said, "All models are wrong; however, some of them are useful." Along the same lines, the DeGroot model is a fairly simple model but it can provide us with useful insights about the learning process in social networks. Indeed, the simplicity of this model makes it tractable for theoretical studies. Specifically, we can analyze different network structure to see for which structures these naive agents can successfully aggregate decentralized information. Since the DeGroot model can be considered a Markov chain, provided that a network is strongly connected (so there is a direct path from any agent to any other) and satisfies a weak aperiodicity condition, beliefs will converge to a consensus. When consensus is reached, the belief of each agent is a weighted average of agents' initial beliefs. These weights provide a measure of social influence.

统计学家乔治 · e · p · 博克斯曾经说过: “所有的模型都是错误的; 然而，其中一些是有用的。”同样的道理，DeGroot 模型是一个相当简单的模型，但它可以为我们提供关于社交网络中学习过程的有用见解。事实上，这个模型的简单性使它易于理论研究。具体来说，我们可以分析不同的网络结构，看看这些天真的代理能够成功地聚合分散的信息。由于 DeGroot 模型可以认为是一个马尔可夫链，只要网络是强连通的(因此从任何主体到任何其他主体都有一个直接的路径)并且满足弱的非周期性条件，信念将收敛到一致意见。当达成共识时，每个代理人的信念就是代理人初始信念的加权平均数。这些权重提供了衡量社会影响力的指标。

In the case of a converging opinion dynamic, the social network is called *wise* if the consensus belief is equal to the true state of the world. It can be shown that the necessary and sufficient condition for wisdom is that the influence of the most influential agent vanishes as the network grows. The speed of convergence is irrelevant to the wisdom of the social network.^{[3]}

In the case of a converging opinion dynamic, the social network is called wise if the consensus belief is equal to the true state of the world. It can be shown that the necessary and sufficient condition for wisdom is that the influence of the most influential agent vanishes as the network grows. The speed of convergence is irrelevant to the wisdom of the social network.

在意见趋同动态的情况下，如果共识信念等于世界的真实状态，那么社会网络就被称为智慧。结果表明，智慧的充要条件是，随着网络的增长，最有影响力的主体的影响力逐渐消失。融合的速度与社交网络的智慧无关。

## Empirical evaluation of models

Along with the theoretical framework for modeling social learning phenomenon, there has been a great amount of empirical research to assess the explanatory power of these models. In one such experiment, 665 subjects in 19 villages in Karnataka, India, were studied while communicating information with each other to learn the true state of the world. This study attempted to distinguish between two most prominent models of information aggregation in social networks, namely, Bayesian learning and DeGroot learning. The study showed that agents' aggregate behavior is statistically significantly better described by the DeGroot learning model.^{[2]}

Along with the theoretical framework for modeling social learning phenomenon, there has been a great amount of empirical research to assess the explanatory power of these models. In one such experiment, 665 subjects in 19 villages in Karnataka, India, were studied while communicating information with each other to learn the true state of the world. This study attempted to distinguish between two most prominent models of information aggregation in social networks, namely, Bayesian learning and DeGroot learning. The study showed that agents' aggregate behavior is statistically significantly better described by the DeGroot learning model.

随着社会学习现象模型的理论框架的建立，已经有大量的实证研究来评估这些模型的解释力。在这样一个实验中，研究人员对 Karnataka 19个村庄的665名受试者进行了研究，他们在互相交流信息的同时，了解了世界的真实状况。本研究试图区分社会网络中两种最主要的信息聚合模型，即贝叶斯学习和 DeGroot 学习。研究表明，DeGroot 学习模型对个体聚集行为的描述具有统计上的显著性。

## References

- ↑
^{1.0}^{1.1}Acemoglu, Daron; Ozdaglar, Asuman (2010). "Opinion Dynamics and Learning in Social Networks".*Dynamic Games and Applications*.**1**(1): 3–49. CiteSeerX 10.1.1.471.6097. doi:10.1007/s13235-010-0004-1. - ↑
^{2.0}^{2.1}Chandrasekhar, Arun G.; Larreguy, Horacio; Xandri, Juan Pablo (August 2015). "Testing Models of Social Learning on Networks: Evidence from a Lab Experiment in the Field".*NBER Working Paper No. 21468*. doi:10.3386/w21468. - ↑ Golub, Benjamin; Jackson, Matthew (2010). "Naïve Learning in Social Networks and the Wisdom of Crowds".
*American Economic Journal: Microeconomics*.**2**(1): 112–149. CiteSeerX 10.1.1.304.7305. doi:10.1257/mic.2.1.112.

Category:Social learning theory Category:Bayesian inference

类别: 社会学习理论类别: 贝叶斯推断

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