In this work, we propose a method of opinion pooling based on pairwise interactions. We assume that each agent has a probability measure on the possible outcomes of some situation, and they try to find a single measure aggregating their estimates. This is a classical problem in Decision Theory, where expert opinions contain some degree of uncertainty, and a Decision Taker needs to pool these estimates. We study this problem using a kinetic theory approach, obtaining a Boltzmann type equation for opinions which are symmetric probability measures defined on the real line. We obtain a non-local, first order, mean field equation as its grazing limit when the parameter in the interaction goes to zero. Also, we prove the convergence to quasi-consensus with explicit estimates on the convergence time depending on the variance of these measures. Let us remark that this model can be interpreted as a noisy model of opinion dynamics. In many models, the opinion of each agent is a point in the real line, the agents interact and observe other agents opinions. We can consider that observed opinions are perturbed or deformed by some noise in the transmission channel or in the interpretation of the agents, so we can think of agents opinions directly as random variables instead of a single point.

中文翻译：

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测量值的意见动态

在这项工作中，我们提出了一种基于成对交互的意见汇集方法。我们假设每个代理对某些情况的可能结果都有一个概率度量，并且他们试图找到一个单一的度量来汇总他们的估计。这是决策理论中的一个经典问题，其中专家意见包含一定程度的不确定性，决策者需要汇总这些估计。我们使用动力学理论方法研究这个问题，获得一个玻尔兹曼型方程，用于在实线上定义的对称概率度量的意见。当相互作用中的参数变为零时，我们获得了一个非局部的一阶平均场方程作为其放牧极限。此外，我们通过根据这些度量的方差对收敛时间进行显式估计来证明收敛到准共识。让我们注意，该模型可以解释为意见动态的嘈杂模型。在许多模型中，每个代理的意见是实线中的一个点，代理交互并观察其他代理的意见。我们可以认为观察到的意见会受到传输通道中或代理解释中的一些噪声的干扰或变形，因此我们可以将代理意见直接视为随机变量而不是单个点。