Approximation Techniques for the Reconstruction of the Probability Measure and the Coupling Parameters in a Curie-Weiss Model for Large Populations

TL;DR

This paper develops a computationally efficient estimator for the coupling parameters in a multi-group Curie-Weiss model, enabling large population analysis without exponential complexity, with applications in social sciences and voting systems.

Contribution

It introduces a novel large population asymptotic approximation method for estimating coupling parameters with low computational cost and proven statistical properties.

Findings

Estimator is consistent for large populations.

Estimator is asymptotically normal.

Estimator satisfies large deviation principles.

Abstract

The Curie-Weiss model, originally used to study phase transitions in statistical mechanics, has been adapted to model phenomena in social sciences where many agents interact with each other. Reconstructing the probability measure of a Curie-Weiss model via the maximum likelihood method runs into the problem of computing the partition function which scales exponentially with the population. We study the estimation of the coupling parameters of a multi-group Curie-Weiss model using large population asymptotic approximations for the relevant moments of the probability distribution in the case that there are no interactions between groups. As a result, we obtain an estimator which can be calculated at a low and constant computational cost for any size of the population. The estimator is consistent (under the added assumption that the population is large enough), asymptotically normal, and satisfies large deviation principles. The estimator is potentially useful in political science, sociology, automated voting, and in any application where the degree of social cohesion in a population has to be identified. The Curie-Weiss model's coupling parameters provide a natural measure of social cohesion. We discuss the problem of estimating the optimal weights in two-tier voting systems.