Inference in Ising Models

The Ising spin glass is a one-parameter exponential family model for binary data with quadratic sufficient statistic. In this paper, we show that given a single realization from this model, the maximum pseudolikelihood estimate (MPLE) of the natural parameter is $\sqrt{a_N}$ consistent at a point whenever the log-partition function has order $a_N$ in a neighborhood of that point. This gives consistency rates of the MPLE for ferromagnetic Ising models on general weighted graphs in all regimes, extending results of Chatterjee (2007) where only $\sqrt N$-consistency of the MPLE was shown. It is also shown that consistent testing, and hence estimation, is impossible in the high temperature phase in ferromagnetic Ising models on a converging sequence of weighted graphs, which includes the Curie-Weiss model. In this regime, the sufficient statistic is distributed as a weighted sum of independent $\chi^2_1$ random variables, and the asymptotic power of the most powerful test is determined.