Reconstructing the Probability Measure of a Curie-Weiss Model Observing the Realisations of a Subset of Spins

We study the problem of reconstructing the probability measure of the Curie-Weiss model from a sample of the voting behaviour of a subset of the population. While originally used to study phase transitions in statistical mechanics, the Curie-Weiss or mean-field model has been applied to study phenomena, where many agents interact with each other. It is useful to measure the degree of social cohesion in social groups, which manifests in the way the members of the group influence each others' decisions. In practice, statisticians often only have access to survey data from a representative subset of a population. As such, it is useful to provide methods to estimate social cohesion from such data. The estimators we study have some positive properties, such as consistency, asymptotic normality, and large deviation principles. The main advantages are that they require only a sample of votes belonging to a (possibly very small) subset of the population and have a low computational cost. Due to the wide application of models such as Curie-Weiss, these estimators are potentially useful in disciplines such as political science, sociology, automated voting, and preference aggregation.