标题： 伊辛模型中的推断 显示英文标题

标题： Inference in Ising Models

摘要： 伊辛自旋玻璃是一个用于二元数据的一参数指数族模型，其充分统计量为二次型。 在本文中，我们证明，给定该模型的一个实现，自然参数的最大伪似然估计（MPLE）在某一点处是$\sqrt {a_N}$-一致的，只要该点邻域内的对数分母函数的阶数为$a_N$。 这给出了在所有情况下一般加权图上的铁磁伊辛模型的MPLE的一致性速率，扩展了Chatterjee（2007）的结果，其中仅展示了MPLE的$\sqrt N$-一致性。 还表明，在收敛简单图序列上的铁磁伊辛模型的高温相中，一致检验以及因此估计是不可能的，这包括Curie--Weiss模型。 在这个区域中，充分统计量分布为独立$\chi^2_1$随机变量的加权和，且最强检验的渐近功效被确定。 我们还展示了我们的结果在合成和现实世界网络数据中的应用。 显示英文摘要

摘要： 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.