Reconstruction of the Probability Measure and the Coupling Parameters in a Curie-Weiss Model

TL;DR

This paper develops a maximum likelihood estimator for the coupling parameters of a multi-group Curie-Weiss model, enabling the reconstruction of the probability measure and quantification of social cohesion from data.

Contribution

It introduces a consistent, asymptotically normal estimator for the model's parameters, addressing practical challenges like partition function computation.

Findings

Estimator is consistent and asymptotically normal.

Probabilities of large deviations decay exponentially.

Applications include social cohesion measurement and voting system analysis.

Abstract

The Curie-Weiss model is used to study phase transitions in statistical mechanics and has been the object of rigorous analysis in mathematical physics. We analyse the problem of reconstructing the probability measure of a multi-group Curie-Weiss model from a sample of data by employing the maximum likelihood estimator for the coupling parameters of the model, under the assumption that there is interaction within each group but not across group boundaries. The estimator has a number of positive properties, such as consistency, asymptotic normality, and exponentially decaying probabilities of large deviations of the estimator with respect to the true parameter value. A shortcoming in practice is the necessity to calculate the partition function of the Curie-Weiss model, which scales exponentially with respect to the population size. There are a number of applications of the estimator in political science, sociology, and automated voting, centred on the idea of identifying the degree of social cohesion in a population. In these applications, the coupling parameter is a natural way to quantify social cohesion. We treat the estimation of the optimal weights in a two-tier voting system, which requires the estimation of the coupling parameter.