• Bernoulli
  • Volume 24, Number 1 (2018), 493-525.

Inference in Ising models

Bhaswar B. Bhattacharya and Sumit Mukherjee

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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 (Ann. Statist. 35 (2007) 1931–1946) 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.

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Bernoulli, Volume 24, Number 1 (2018), 493-525.

Received: November 2015
Revised: July 2016
First available in Project Euclid: 27 July 2017

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exponential family graph limit theory hypothesis testing Ising model pseudolikelihood estimation spin glass


Bhattacharya, Bhaswar B.; Mukherjee, Sumit. Inference in Ising models. Bernoulli 24 (2018), no. 1, 493--525. doi:10.3150/16-BEJ886.

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