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