Joint estimation of parameters in Ising model

We study joint estimation of the inverse temperature and magnetization parameters (β ,B) of an Ising model with a nonnegative coupling matrix Aₙ of size n× 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 Aₙ is the adjacency matrix of a bounded degree graph. If Aₙ is the scaled adjacency matrix of a graph whose average degree goes to +∞, 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.