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.

本文证明了针对一类二元数据的 Ising 自旋玻璃模型，如给定模型的单个实现，其最大伪似然估计值在某一点处是 squraat {a_N}- 一致的，推广了 Chatterjee（2007）的结果，同时在简单图的收敛序列中，证明了在高温相中不可能进行一致的测试和估计。我们还展示了我们的结果在合成和真实世界网络数据上的应用。

Ising 模型中的推断

Inference in Ising Models

We study the existence of a spin-glass phase in a field using Monte Carlo
simulations performed along a nontrivial path in the field--temperature plane
that must cross any putative de Almeida-Thouless instability line. The method
is first tested on the Ising spin glass on a Bethe lattice where the
instability line separating the spin glass from the paramagnetic state is also
computed analytically. While the instability line is reproduced by our
simulations on the mean-field Bethe lattice, no such instability line can be
found numerically for the short-range three-dimensional model.

我们使用蒙特卡罗模拟法沿着场 - 温度平面上的非平凡路径研究自旋玻璃相的存在性，该路径必须穿过任何假定的 de Almeida-Thouless 不稳定性线。该方法首先在 Bethe 格子上的 Ising 自旋玻璃上进行测试，在此模型中，分离自旋玻璃和顺磁态的不稳定线也被解析地计算。虽然在均匀场 Bethe 格子的模拟中，我们的模拟重现了不稳定线，但在三维短程模型中没有找到任何此类不稳定性线数值。

在磁场中的伊辛自旋玻璃的行为

Behavior of Ising Spin Glasses in a Magnetic Field

The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of
neural networks and the Ising spin glass are all models of binary data
belonging to the one-parameter exponential family with quadratic sufficient
statistic. Under bare minimal conditions, we establish the
$\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural
parameter in this family, even at critical temperatures. Since very little is
known about the low and critical temperature regimes of these extremely
difficult models, the proof requires several new ideas. The author's version of
Stein's method is a particularly useful tool. We aim to introduce these
techniques into the realm of mathematical statistics through an example and
present some open questions.

通过作者版本的 Stein 方法，我们建立了一种二进制数据模型的最大伪似然估计方法的根号 N 一致性，并且在关键温度下仍然有效，从而增强了这些模型的数学统计分析。