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Inference for general Ising models

Published online by Cambridge University Press:  14 July 2016

David K. Pickard
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Abstract

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models.

In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

Keywords

ISING MODELSGIBBS ENSEMBLESAUTO-LOGISTIC MODELSNEAREST-NEIGHBOR MODELSMARKOV FIELDSLATTICE MODELSCENTRAL LIMIT THEOREMASYMPTOTIC INFERENCECODING ESTIMATORS
Type
Part 6 — Stochastic Processes
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 345 - 357
Copyright
Copyright © 1982 Applied Probability Trust 

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References

Baxter, R. J. (1972) Partition function of the eight vertex lattice model. Ann. Phys. 70, 193–228.Google Scholar
Besag, J. E. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B36, 192–225.Google Scholar
Besag, J. E. (1977) Efficiency of pseudolikelihood estimation for simple Gaussian fields. Biometrika 64, 616–618.Google Scholar
Besag, J. E. and Moran, P. A. P. (1975) On the estimation and testing of spatial interaction in Gaussian lattice processes. Biometrika 62, 555–562.Google Scholar
Bloemena, A. R. (1964) Sampling From a Graph. Mathematical Center Tracts No. 2, Amsterdam.Google Scholar
Del Grosso, G. (1974) On the local central limit theorem for Gibbs processes. Commun. Math. Phys. 37, 141–160.Google Scholar
Dobrushin, R. L. (1965) Existence of a phase transition in two dimensional and three dimensional Ising lattices. Theory Prob. Appl. 10, 193–213.CrossRefGoogle Scholar
Dobrushin, R. L. (1968a) The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Functional Anal. Appl. 2, 44–57.Google Scholar
Dobrushin, R. L. (1968b) The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Prob. Appl. 13, 197–224.CrossRefGoogle Scholar
Dobrushin, R. L. (1968C) Gibbsian random fields for lattice systems with pairwise interactions. Functional Anal. Appl. 2, 31–43.Google Scholar
Dobrushin, R. L. (1969) Gibbsian random fields. The general case. Functional Anal. Appl. 3, 27–35.CrossRefGoogle Scholar
Gallavotti, G. and Martin Löf, A. (1975) Block spin distributions for short range attractive Ising models. Nuovo Cimento 25, 425–441.CrossRefGoogle Scholar
Gaunt, D. S. (1974) Generalized codes and their application to Ising models with four-spin interactions including eight-vertex model. J. Phys. A 7, 1596–1612.Google Scholar
Hurst, D. A. and Green, H. S. (1960) New solution to the Ising problem for a rectangular lattice. J. Chem. Phys. 33, 1059–1062.CrossRefGoogle Scholar
Kryscio, R. J., Saunders, R. and Funk, G. M. (1980) Normal approximations for binary lattice systems. J. Appl. Prob. 17, 674–685.Google Scholar
Martin Löf, A. (1973) Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Commun. Math. Phys. 32, 75–92.Google Scholar
Mccoy, B. M. and Wu, T. T. (1973) The Two Dimensional Ising Model. Harvard University Press, Cambridge, MA.CrossRefGoogle Scholar
Minlos, R. A. (1967a) Limiting Gibbs distributions. Functional Anal. Appl. 1, 140–150.CrossRefGoogle Scholar
Minlos, R. A. (1967b) Regularity of the Gibbs limit distribution. Functional Anal. Appl. 1, 206–217.Google Scholar
Moran, P. A. P. (1947) Random associations on a lattice. Proc. Camb. Phil. Soc. 43, 321–328.Google Scholar
Moran, P. A. P. (1948) The interpretation of statistical maps. J. R. Statist. Soc. B10, 243–251.Google Scholar
Moran, P. A. P. (1973a) A Gaussian Markovian process on a square lattice. J. Appl. Prob. 10, 54–62.Google Scholar
Moran, P. A. P. (1973b) Necessary conditions for Markovian processes on a lattice. J. Appl. Prob. 10, 605–612.CrossRefGoogle Scholar
Moussouris, J. (1974) Gibbs and Markov random systems with constraints. J. Statist. Phys. 10, 11–33.CrossRefGoogle Scholar
Newell, G. F. and Montroll, ?. W. (1953) On the theory of the Ising model of ferromagnetism. Rev. Mod. Phys. 25, 353–389.Google Scholar
Onsager, L. (1944) Crystal Statistics I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149.Google Scholar
Pickard, D. K. (1976) Asymptotic inference for an Ising lattice. J. Appl. Prob. 13, 486–497.Google Scholar
Pickard, D. K. (1977a) Asymptotic inference for an Ising lattice II. Adv. Appl. Prob. 9, 476–501.Google Scholar
Pickard, D. K. (1977b) A curious binary lattice process. J. Appl. Prob. 14, 717–731.Google Scholar
Pickard, D. K. (1978) Unilateral Ising models. Suppl. Adv. Appl. Prob. 10, 58–64.CrossRefGoogle Scholar
Pickard, D. K. (1979) Asymptotic inference for an Ising lattice III. J. Appl. Prob. 16, 12–24.Google Scholar
Pickard, D. K. (1980) Unilateral Markov fields. Adv. Appl. Prob. 12, 655–671.Google Scholar
Pickard, D. K. (1981) Asymptotic inference for an Ising lattice IV: Besag's coding method.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications. Wiley, New York.CrossRefGoogle Scholar
Ruelle, D. (1969) Statistical Mechanics. Benjamin, New York.Google Scholar
Silvey, S. D. (1970) Statistical Inference. Penguin, Harmondsworth.Google Scholar
Strauss, D. J. (1975) Analyzing binary lattice data with the nearest-neighbor property. J. Appl. Prob. 12, 702–712.CrossRefGoogle Scholar
Strauss, D. J. (1977) Clustering on colored lattices. J. Appl. Prob. 14, 135–143.Google Scholar