Joint estimation of parameters in Ising model

Abstract

We study joint estimation of the inverse temperature and magnetization parameters $(\beta ,B)$ of an Ising model with a nonnegative coupling matrix $A_{n}$ of size $n\times n$, given one sample from the Ising model. We give a general bound on the rate of consistency of the bi-variate pseudo-likelihood estimator. Using this, we show that estimation at rate $n^{-1/2}$ is always possible if $A_{n}$ is the adjacency matrix of a bounded degree graph. If $A_{n}$ is the scaled adjacency matrix of a graph whose average degree goes to $+\infty $, the situation is a bit more delicate. In this case, estimation at rate $n^{-1/2}$ is still possible if the graph is not regular (in an asymptotic sense). Finally, we show that consistent estimation of both parameters is impossible if the graph is Erdős–Renyi with parameter $p>0$ independent of $n$, thus confirming that estimation is harder on approximately regular graphs with large degree.