Joint estimation of parameters in Ising model

We study joint estimation of the inverse temperature and magnetization parameters (.,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 Erd.s.Renyi with parameter p>0 independent of n, thus confirming that estimation is harder on approximately regular graphs with large degree.