In models of opinion dynamics, many parameters -- either in the form of constants or in the form of functions -- play a critical role in describing, calibrating, and forecasting how opinions change with time. When examining a model of opinion dynamics, it is beneficial to infer its parameters using empirical data. In this paper, we study an example of such an inference problem. We consider a mean-field bounded-confidence model with an unknown interaction kernel between individuals. This interaction kernel encodes how individuals with different opinions interact and affect each other's opinions. Because it is often difficult to quantitatively measure opinions as empirical data from observations or experiments, we assume that the available data takes the form of partial observations of a cumulative distribution function of opinions. We prove that certain measurements guarantee a precise and unique inference of the interaction kernel and propose a numerical method to reconstruct an interaction kernel from a limited number of data points. Our numerical results suggest that the error of the inferred interaction kernel decays exponentially as we strategically enlarge the data set.

翻译：在意见动力学模型中，许多参数（无论是常数形式还是函数形式）在描述、校准和预测意见随时间变化方面起着关键作用。在研究意见动力学模型时，利用经验数据推断其参数是有益的。本文研究了此类推断问题的一个实例。我们考虑一个具有未知个体间交互核的平均场有界置信模型。该交互核编码了持有不同意见的个体如何相互作用并影响彼此的意见。由于通过观测或实验以经验数据形式定量测量意见通常较为困难，我们假设可用数据以意见累积分布函数的部分观测形式呈现。我们证明了特定测量条件可保证交互核的精确唯一推断，并提出了一种从有限数据点重建交互核的数值方法。数值结果表明，通过策略性地扩大数据集，推断出的交互核误差呈指数级衰减。