JOINT ESTIMATION OF PARAMETERS IN ISING MODEL

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 x 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 +infinity, 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 Erdos-Renyi with parameter p > 0 independent of n, thus confirming that estimation is harder on approximately regular graphs with large degree.