Joint Estimation in Potts Model

In this paper, we study estimation of parameters in a two-parameter Potts model with $q$ colors and coupling matrix $A_N$. We characterize concrete sufficient conditions for existence of the pseudo-likelihood estimator of the Potts model, in terms of the local magnetic fields, and give sufficient conditions for the validity of the above characterization. We then provide sufficient criteria for estimation of both parameters at the optimal rate $\sqrt{N}$. In particular, if $A_N$ is the scaled adjacency matrix of a graph $G_N$, then we show that joint estimation is possible if either $G_N$ has bounded degree or is irregular. In contrast, we give an example of a graph sequence $G_N$ which is approximately regular and dense, where no consistent estimator exists. We also show that one-parameter estimation at the optimal rate $\sqrt{N}$ holds under much milder conditions when the other parameter is known. Along the way, we develop a concentration result for mean-field Potts models using the framework of nonlinear large deviations. Compared to the Ising case, our results for the Potts case require a novel analysis across multiple colors.