Consistency of Maximum Likelihood and Pseudo-Likelihood Estimators for Gibbs Distributions

  • B. Gidas
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 10)


We prove that the Maximum Likelihood and Pseudo-likelihood estimators for the parameters of Gibbs distributions (equivalently Markov Random Fields) over ℤd, d≥l, are consistent even at points of “first” or “higher-order” phase transitions. The distributions are parametrized by points in a finite-dimensional Euclidean space ℝm, m≥l, and the single spin state space is either a finite set or a compact metric space. Also, the underlying interactions need not be of finite range.


Asymptotic Normality Perceptual Inference Single Photon Emission Tomography Strict Convexity Gibbs Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Besag, J.: “On the Statistical Analysis of Dirty Pictures”, to appear in J. Roy. Stat. Society, series B, 1986.Google Scholar
  2. 2.
    Besag, J.: “Spatial Interaction and Statistical Analysis of Lattice Systems”, J. Roy. Stat. Society, series B, 36 (1974) 192–236.MathSciNetMATHGoogle Scholar
  3. 3.
    Besag, J.: “Statistical Analysis of Non-Lattice Data”, The Statistician 24 (1975) 179–195.CrossRefGoogle Scholar
  4. 4.
    Besag, J.: “Efficiency of Pseudo-likelihood Estimation for Simple Gaussian Fields”, Biometrika 64 (1977) 616–618.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chellappa, R., and R.L. Kashyap: “Digital Image Restoration using Interaction Models”, IEEE Trans. Acoust., Speech, Signal Processing 30 (1982) 461–472.CrossRefGoogle Scholar
  6. 6.
    Cohen, F.S., and D.B. Cooper: “Simple Parallel Hierarchical and Relaxation Algorithms for Segmenting Noncausal Markovian Random Fields”, IEEE PAMI, to appear.Google Scholar
  7. 7.
    Cross, G.R., and A.K. Jain: “Markov Random Field Texture Models”, IEEE PAMI 5 (1983) 25–40.CrossRefGoogle Scholar
  8. 8.
    Elliott, H., and H. Darin: “Modeling and Segmentation of Noisy and Textured Images using Gibbs Random Fields”, IEEE PAMI, to appear.Google Scholar
  9. 9.
    Geman, S., and D. Geman: “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”, IEEE PAMI 6 (1984) 721–741.MATHCrossRefGoogle Scholar
  10. 10.
    Geman, D., and S. Geman: “Bayesian Image Analysis”, in Disordered Systems and Biological Organization, Springer-Verlag (1986), eds: E. Bienenstock, et. al.Google Scholar
  11. 11.
    Geman, D., and S. Geman: “Parameter Estimation for Some Markov Random Fields”, Brown Univ. Tech. Rep. No. 11, August 1983.Google Scholar
  12. 12.
    Geman, D., S. Geman, and D.E. McClure: in preparation (invited paper, Annals of Statistics).Google Scholar
  13. 13.
    Geman, S., and C. Graffine: “Markov Random Field Image Models and their Applications to Computer Vision”, Proceedings of the 1986 International Congress of Mathematicians. Ed. A.M. Gleason, AMS, New York, 1987.Google Scholar
  14. 14.
    Geman, S., and D.E. McClure: “Bayesian Image Analysis: An Application to Single Photon Emission Tomography”, 1985 Proceedings of the American Statistical Association. Statistical Computing Section.Google Scholar
  15. 15.
    Gidas, B.: “A Renormalization Group Approach to Image Processing Problems” preprint 1986.Google Scholar
  16. 16.
    Gidas, B.: “Parameter Estimation for Gibbs Distributions”, preprint, November 1986.Google Scholar
  17. 17.
    Gidas, B.: “Asymptotic Normality and Efficiency of Pseudo-likelihood Estimators”, in preparation.Google Scholar
  18. 18.
    Graffine, C.: Thesis, Division of Applied Mathematics, Brown University, 1987.Google Scholar
  19. 19.
    Grenander, U.: Tutorial in Pattern Theory, Division of Applied Mathematics, Brown University, 1983.Google Scholar
  20. 20.
    Guyon, X.: “Estimation d’un champ par pseudo-rraisemblaine conditionelle: etude asymptotique et application au cas Markovien”, to appear in Actes de la 6eme Recontre Franco-Belge de Staticiens, Bruxelles, Nov. 1985.Google Scholar
  21. 21.
    Hinton, G.E., and T.J. Sejnowski: “Optimal Perceptual Inference”, in Proceedings IEEE Conference Computer Vision Pattern Recognition 1983.Google Scholar
  22. 22.
    Hopfield, J.: “Neural Networks and Physical Systems With Emergent Collective Computational Abilities”, in Proceedings of the National Academy of Sciences, U.S.A., Vol. 79 (1982), 2554–2558.MathSciNetCrossRefGoogle Scholar
  23. 23.
    von der Malsburg, C., and E. Bienenstock: “Statistical Coding and Short Term Synaptic Plasticity: A Scheme for Knowledge Representation in the Brain”, in Disordered Systems and Biological Organization, Springer-Verlag 1986, eds.: E. Bienenstock, Scholar
  24. 24.
    Marroquin, J., S. Mitter, and T. Poggio: “Probabilistic Solution of Ill-posed Problems in Computational Vision”, Art. Intell. Lab., Tech. Report, M.I.T. 1985.Google Scholar
  25. 25.
    Passolo, A.: “Estimation of Binary Markov Random Fields”, preprint, Department of Statistics, University of Washington, Seattle (1986).Google Scholar
  26. 26.
    Pickard, D.K.: “Asymptotic Inference for Ising Lattice III. Non-zero Field and Ferromagnetic States”, J. Appl. Prob. 16 (1979) 12–24.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Ripley, B.D.: “Statistics, Images, and Pattern Recognition”, The Canadian Jour. of Stat. 14 (1986) 83–111.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Ruelle, D.: Thermodynamic Formalism, Addison-Wesley, 1978.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • B. Gidas
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations