The Annals of Statistics

Estimation in spin glasses: A first step

Sourav Chatterjee

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The Sherrington–Kirkpatrick model of spin glasses, the Hopfield model of neural networks and the Ising spin glass are all models of binary data belonging to the one-parameter exponential family with quadratic sufficient statistic. Under bare minimal conditions, we establish the $\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural parameter in this family, even at critical temperatures. Since very little is known about the low and critical temperature regimes of these extremely difficult models, the proof requires several new ideas. The author’s version of Stein’s method is a particularly useful tool. We aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions.

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Ann. Statist., Volume 35, Number 5 (2007), 1931-1946.

First available in Project Euclid: 7 November 2007

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Zentralblatt MATH identifier

Primary: 62F10: Point estimation 62F12: Asymptotic properties of estimators 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Spin glass neural networks estimation consistency exponential families


Chatterjee, Sourav. Estimation in spin glasses: A first step. Ann. Statist. 35 (2007), no. 5, 1931--1946. doi:10.1214/009053607000000109.

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  • Aizenman, M., Lebowitz, J. L. and Ruelle, D. (1987). Some rigorous results on the Sherrington–Kirkpatrick spin glass model. Comm. Math. Phys. 112 3–20.
  • Aizenman, M., Sims, R. and Starr, S. (2003). Extended variational principle for the Sherrington–Kirkpatrick spin-glass model. Phys. Rev. B 68 214403.
  • Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review (with discussion). Statist. Sinica 9 611–677.
  • Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192–236.
  • Besag, J. (1975). Statistical analysis of non-lattice data. Statistician 24 179–195.
  • Bovier, A. and Gayrard, V. (1998). Hopfield models as generalized random mean field models. In Mathematical Aspects of Spin Glasses and Neural Networks (A. Bovier and P. Picco, eds.) 3–89. Birkhäuser, Boston.
  • Chatterjee, S. (2005). Concentration inequalities with exchangeable pairs. Ph.D. dissertation, Stanford Univ. Available at
  • Chatterjee, S. (2007). Concentration of Haar measures, with an application to random matrices. J. Funct. Anal. 245 379–389.
  • Chatterjee, S. (2007). Stein's method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
  • 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.
  • Comets, F. and Neveu, J. (1995). The Sherrington–Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case. Comm. Math. Phys. 166 549–564.
  • Geyer, C. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data (with discussion). J. Roy. Statist. Soc. Ser. B 54 657–699.
  • Gidas, B. (1988). Consistency of maximum likelihood and pseudolikelihood estimators for Gibbs distributions. In Stochastic Differential Systems, Stochastic Control Theory and Applications (W. Fleming and P.-L. Lions, eds.) 129–145. Springer, New York.
  • Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
  • Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
  • Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13–20.
  • Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. U.S.A. 79 2554–2558.
  • Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Z. Physik 31 253–258.
  • Jensen, J. L. and Møller, J. (1991). Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1 445–461.
  • Jerrum, M. and Sinclair, A. (1993). Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22 1087–1116.
  • Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. (2) 65 117–149.
  • Panchenko, D. (2005). Free energy in the generalized Sherrington–Kirkpatrick mean field model. Rev. Math. Phys. 17 793–857.
  • Shcherbina, M. V. (1997). On the replica symmetric solution for the Sherrington–Kirkpatrick model. Helv. Phys. Acta 70 838–853.
  • Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin-glass. Phys. Rev. Lett. 35 1792–1796.
  • Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin.
  • Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.