  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 the 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
simple graphs, which include 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. We also illustrate applications of our results on synthetic and
real-world network data.
