Social Learning is a decentralized decision-making paradigm in which spatially dispersed agents collect streaming observations regulated by one of a finite number of models (the hypotheses). The agents are interested in assigning probability scores (the beliefs) to the possible hypotheses. To this end, the agents exchange their beliefs according to a certain communication graph. It has been shown that, under reasonable conditions on the identifiability of the decision model and the network connectivity, each agent ultimately places all the belief mass on the true hypothesis governing the data. However, several questions remain unanswered regarding the evaluation of the social learning performance. One recently adopted performance metric is the rejection rate, i.e., the rate at which the beliefs about the erroneous hypotheses vanish. One contribution of this work is to establish that the rejection rate leads to several paradoxes, which make it unsuitable as a valid performance measure. We then focus on studying the error probability measure. For a binary Gaussian problem, we derive an analytical formula characterizing the ratio between the individual agents' probabilities and the optimal Bayesian probability. The formula shows that this ratio is expressed by the product of two terms quantifying the effect of the network connectivity and the role of the prior information. As a result, an irreducible gap emerges between the decentralized and the centralized error probabilities, which is agent-dependent and does not disappear asymptotically.

翻译：社会学习是一种去中心化决策范式，其中空间分布的主体收集由有限数量模型（即假设）之一调控的流式观测数据。主体旨在为可能的假设分配概率评分（即信念）。为此，主体根据特定通信图交换信念。已有研究表明，在决策模型可辨识性和网络连通性的合理条件下，每个主体最终会将全部信念质量集中于支配数据的真实假设。然而，关于社会学习性能评估的若干问题仍未解答。近期采用的一种性能指标是拒绝率，即关于错误假设的信念消失的速率。本文的贡献之一是证明拒绝率会导致若干悖论，使其不适合作为有效的性能度量。随后，我们重点研究错误概率度量。针对二元高斯问题，我们推导出一个解析公式，刻画个体主体概率与最优贝叶斯概率之间的比率。该公式表明，该比率由量化网络连通性效应和先验信息作用的两项乘积表示。因此，去中心化与集中化错误概率之间出现了一个不可消除的差距，该差距依赖于主体且不会渐近消失。