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Inference in Ising models

Bhaswar B. Bhattacharya, Sumit Mukherjee
2018 Bernoulli  
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 √ 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
more » ... regimes, extending the results of Chatterjee (Ann. Statist. 35 (2007Statist. 35 ( ) 1931Statist. 35 ( -1946 where only √ 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 χ 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.
doi:10.3150/16-bej886 fatcat:olyawwghfzeknhz352fpd6jude