## Bernoulli

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

### Inference in Ising models

#### 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
Revised: July 2016
First available in Project Euclid: 27 July 2017

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

Digital Object Identifier
doi:10.3150/16-BEJ886

Mathematical Reviews number (MathSciNet)
MR3706767

Zentralblatt MATH identifier
06778338

#### 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

#### References

• [1] Anandkumar, A., Tan, V.Y.F., Huang, F. and Willsky, A.S. (2012). High-dimensional structure estimation in Ising models: Local separation criterion. Ann. Statist. 40 1346–1375.
• [2] Banerjee, S., Carlin, B.P. and Gelfand, A.E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Boca Raton, FL: Chapman & Hall.
• [3] Basak, A. and Mukherjee, S. (2016). Universality of the mean-field for the Potts model. Probab. Theory Related Fields. To appear. Available at arXiv:1508.03949.
• [4] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B. Stat. Methodol. 36 192–236.
• [5] Besag, J. (1975). Statistical analysis of non-lattice data. Statistician 24 179–195.
• [6] Bhattacharya, B.B., Diaconis, P. and Mukherjee, S. (2016). Universal Poisson and normal limit theorems in graph coloring problems with connections to extremal combinatorics. Ann. Appl. Probab. To appear. Available at arXiv:1310.2336.
• [7] Borgs, C., Chayes, J.T., Lovász, L., Sós, V.T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
• [8] Borgs, C., Chayes, J.T., Lovász, L., Sós, V.T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
• [9] Bresler, G. (2015). Efficiently learning Ising models on arbitrary graphs [extended abstract]. In STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing 771–782. New York: ACM.
• [10] Chatterjee, S. (2007). Estimation in spin glasses: A first step. Ann. Statist. 35 1931–1946.
• [11] Comets, F. (1992). On consistency of a class of estimators for exponential families of Markov random fields on the lattice. Ann. Statist. 20 455–468.
• [12] Comets, F. and Gidas, B. (1991). Asymptotics of maximum likelihood estimators for the Curie–Weiss model. Ann. Statist. 19 557–578.
• [13] Dembo, A. and Montanari, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137–211.
• [14] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565–592.
• [15] Dembo, A., Montanari, A. and Sun, N. (2013). Factor models on locally tree-like graphs. Ann. Probab. 41 4162–4213.
• [16] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659–1664.
• [17] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
• [18] Geyer, C.J. and Thompson, E.A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 54 657–699.
• [19] Gidas, B. (1988). Consistency of maximum likelihood and pseudolikelihood estimators for Gibbs distributions. In Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, MN, 1986) (M. Fleming and P.-L. Lions, eds.). IMA Vol. Math. Appl. 10 129–145. New York: Springer.
• [20] Green, P.J. and Richardson, S. (2002). Hidden Markov models and disease mapping. J. Amer. Statist. Assoc. 97 1055–1070.
• [21] Guyon, X. and Künsch, H.R. (1992). Asymptotic comparison of estimators in the Ising model. In Stochastic Models, Statistical Methods, and Algorithms in Image Analysis (Rome, 1990). Lecture Notes in Statist. 74 177–198. Berlin: Springer.
• [22] Hopfield, J.J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79 2554–2558.
• [23] Ising, E. (1925). Beitrag zur theorie der ferromagnetismus. Zeitschrift Für Physik 31 253–258.
• [24] Jensen, J.L. and Møller, J. (1991). Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1 445–461.
• [25] Krivelevich, M. and Sudakov, B. (2003). The largest eigenvalue of sparse random graphs. Combin. Probab. Comput. 12 61–72.
• [26] Levin, D.A., Luczak, M.J. and Peres, Y. (2010). Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability. Probab. Theory Related Fields 146 223–265.
• [27] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Providence, RI: Amer. Math. Soc.
• [28] Lubetzky, E. and Sly, A. (2013). Cutoff for the Ising model on the lattice. Invent. Math. 191 719–755.
• [29] Lubetzky, E. and Sly, A. (2016). Information percolation and cutoff for the stochastic Ising model. J. Amer. Math. Soc. 29 729–774.
• [30] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer Monographs in Mathematics. New York: Springer.
• [31] Pickard, D.K. (1987). Inference for discrete Markov fields: The simplest nontrivial case. J. Amer. Statist. Assoc. 82 90–96.
• [32] Ravikumar, P., Wainwright, M.J. and Lafferty, J.D. (2010). High-dimensional Ising model selection using $\ell_{1}$-regularized logistic regression. Ann. Statist. 38 1287–1319.
• [33] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume I: Basic Examples. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 54. Berlin: Springer.
• [34] Talagrand, M. (2011). Mean Field Models for Spin Glasses: Advanced Replica-Symmetry and Low Temperature. Volume II. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 55. Heidelberg: Springer.
• [35] Xue, L., Zou, H. and Cai, T. (2012). Nonconcave penalized composite conditional likelihood estimation of sparse Ising models. Ann. Statist. 40 1403–1429.