Blume-Capel model: Estimation of a three stable state network for $-\bf 1$, $\bf 0$ and $\bf +1$ data

TL;DR

This paper introduces the Blume-Capel model as an extension of the Ising model for data with three states, demonstrating its estimation via pseudo-likelihood and lasso methods, with applications to voting preference data.

Contribution

The paper develops a parameter estimation approach for the BC model, including its identification and accurate recovery in small networks, using pseudo-likelihood and regularization techniques.

Findings

The BC model has three stable states and is part of the exponential family.

Pseudo-likelihood combined with lasso accurately estimates BC parameters.

Confidence intervals with good coverage are obtained using desparsified lasso methods.

Abstract

An extension of the Ising model is proposed as a viable alternative for data with values $-1$, $0$ and $+1$ in the inverse problem, i.e., estimation of the parameters. This model is called the Blume-Capel (BC) model, adapted from physics for small networks. The advantage of the BC model is not only the fact that it is possible to have a neutral (centrist) position on the response scale, but also that this model allows for three stable states. We illustrate magnetisation properties of the BC model using simulations and mean field results. For estimation of the BC parameters, we show that the BC model is part of the exponential family of distributions and show that the model is identified, except for the (inverse) temperature. We then show that combining pseudo-likelihood with lasso yields accurate parameter recovery for the BC model, even in small networks. Moreover, confidence intervals with good coverage properties can be obtained using the desparsified lasso together with sandwich and shrinkage techniques. We apply the methods to data obtained from the online platform \textit{Stemwijzer}, intended to aid people in deciding for whom to vote.