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 non-negative 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 pseudolikelihood 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 Erd\"os-Renyi with parameter $p>0$ free of $n$, thus confirming that estimation is harder on approximately regular graphs with large degree.