Reconstruction of the Probability Measure and the Coupling Parameters in a Curie-Weiss Model
Miguel Ballesteros, Ramsés H. Mena, Arno Siri-Jégousse, Gabor Toth
TL;DR
This work develops a rigorous statistical framework for reconstructing the probability measure and estimating the coupling parameters of a multi-group Curie-Weiss model using maximum likelihood, under intra-group interactions and no cross-group coupling. The authors define a sufficient statistic $\boldsymbol{T}$ and an MLE $\hat{\boldsymbol{\beta}}_{\boldsymbol{N}}$ via the implicit condition $\mathbb{E}_{\hat{\boldsymbol{\beta}}_{\boldsymbol{N}},\boldsymbol{N}}(S_{1}^{2},\ldots,S_{M}^{2})=\boldsymbol{T}(x)$, and establish estimator well-definedness, consistency, asymptotic normality, and a large-deviation principle, while noting the partition-function cost. They analyze the standard error of the margin statistic $T$, derive a CLT, and obtain a delta-method-based propagation of uncertainty to $\hat{\boldsymbol{\beta}}_{\boldsymbol{N}}$. As an application, they show how to derive optimal weights in a two-tier voting system from the estimated coupling parameters, showing $w_{\lambda}=\mathbb{E}_{\beta_{\lambda},N_{\lambda}}|S_{\lambda}|$, and provide plug-in estimators with corresponding consistency, asymptotic normality, and LDP properties. Overall, the paper bridges statistical physics, inference, and political science by quantifying social cohesion and informing voting-weights design through principled probabilistic estimation.
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.