## Bernoulli

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

### Inference in Ising models

Bhaswar B. Bhattacharya and Sumit Mukherjee

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

Received: November 2015

Revised: July 2016

First available in Project Euclid: 27 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1501142453

**Digital Object Identifier**

doi:10.3150/16-BEJ886

**Mathematical Reviews number (MathSciNet)**

MR3706767

**Zentralblatt MATH identifier**

06778338

**Keywords**

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

#### Citation

Bhattacharya, Bhaswar B.; Mukherjee, Sumit. Inference in Ising models. Bernoulli 24 (2018), no. 1, 493--525. doi:10.3150/16-BEJ886. https://projecteuclid.org/euclid.bj/1501142453