Statistics and Computing

, Volume 27, Issue 5, pp 1271–1292 | Cite as

Approximate computations for binary Markov random fields and their use in Bayesian models

Article

Abstract

Discrete Markov random fields form a natural class of models to represent images and spatial datasets. The use of such models is, however, hampered by a computationally intractable normalising constant. This makes parameter estimation and a fully Bayesian treatment of discrete Markov random fields difficult. We apply approximation theory for pseudo-Boolean functions to binary Markov random fields and construct approximations and upper and lower bounds for the associated computationally intractable normalising constant. As a by-product of this process we also get a partially ordered Markov model approximation of the binary Markov random field. We present numerical examples with both the pairwise interaction Ising model and with higher-order interaction models, showing the quality of our approximations and bounds. We also present simulation examples and one real data example demonstrating how the approximations and bounds can be applied for parameter estimation and to handle a fully Bayesian model computationally.

Keywords

Approximate inference Bayesian analysis Discrete Markov random fields Image analysis Pseudo-Boolean functions Spatial data Variable elimination algorithm 

References

  1. Austad, H.M.: Approximations of binary Markov random fields. PhD thesis, Norwegian University of Science and Technology. Thesis number 292:2011. Available from http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-14922 (2011)
  2. Besag, J.: Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36, 192–225 (1974)MathSciNetMATHGoogle Scholar
  3. Besag, J.: On the statistical analysis of dirty pictures (with discussion). J. R. Stat. Soc. Ser. B 48, 259–302 (1986)MATHGoogle Scholar
  4. Clifford, P.: Markov random fields in statistics. In: Grimmett, G.R., Welsh, D.J.A. (eds.) Disorder in Physical Systems, pp. 19–31. Oxford University Press (1990)Google Scholar
  5. Cowell, R.G., Dawid, A.P., Lauritzen, S.L., Spiegelhalter, D.J.: Probabilistic Networks and Expert Systems, Exact Computational Methods for Bayesian Networks. Springer, London (2007)MATHGoogle Scholar
  6. Cressie, N.A.C.: Statistics for Spatial Data, 2nd edn. Wiley, New York (1993)MATHGoogle Scholar
  7. Cressie, N., Davidson, J.: Image analysis with partially ordered Markov models. Comput. Stat. Data Anal. 29, 1–26 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. Ding, G., Lax, R., Chen, J., Chen, P.P.: Formulas for approximating pseudo-Boolean random variables. Discret. Appl. Math. 156, 1581–1597 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. Ding, G., Lax, R., Chen, J., Chen, P.P., Marx, B.D.: Transforms of pseudo-Boolean random variables. Discret. Appl. Math. 158, 13–24 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. Friel, N., Rue, H.: Recursive computing and simulation-free inference for general factorizable models. Biometrika 94, 661–672 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. Friel, N., Pettitt, A.N., Reeves, R., Wit, E.: Bayesian inference in hidden Markov random fields for binary data defined on large lattices. J. Comput. Graph. Stat. 18, 243–261 (2009)MathSciNetCrossRefGoogle Scholar
  12. Gelman, A., Meng, X.-L.: Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Stat. Sci. 13, 163–185 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. Geyer, C.J., Thompson, E.A.: Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Am. Stat. Assoc. 90, 909–920 (1995)CrossRefMATHGoogle Scholar
  14. Grabisch, M., Marichal, J.L., Roubens, M.: Equivalent representations of set functions. Math. Oper. Res. 25, 157–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. Green, P.J.: Reversible jump MCMC computation and Bayesian model determination. Biometrika 82, 711–732 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. Grelaud, A., Robert, C., Marin, J.M., Rodolphe, F., Taly, J.F.: ABC likelihood-free methods for model choice in Gibbs random fields. Bayesian Anal. 4, 317–336 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. Gu, M.G., Zhu, H.T.: Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation. J. R. Stat. Soc. Ser. B 63, 339–355 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. Hammer, P.L., Holzman, R.: Approximations of pseudo-Boolean functions; applications to game theory. Methods Models Oper. Res. 36, 3–21 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. Hammer, P.L., Rudeanu, S.: Boolean Methods in Operation Research and Related Areas. Springer, Berlin (1968)CrossRefMATHGoogle Scholar
  20. Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22, 1087–1116 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. Künsch, H.R.: State space and hidden Markov models. In: Barndorff-Nielsen, O.E., Cox, D.R., Klppelberg, C. (eds.) Complex Stochastic Systems. Chapman & Hall/CRC (2001)Google Scholar
  22. Liang, F.: A double Metropolis-Hastings sampler for spatial models with intractable normalizing constants. J. Stat. Comput. Simul. 80, 1007–1022 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. Liang, F., Liu, C., Carroll, R.: Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley, New York (2011)MATHGoogle Scholar
  24. Lyne, A.M., Girolami, M., Atchadé, Y., Strathmann, H., Simplson, D.: On Russian roulette estimates for Bayesian inference with doubly-intractable likelihoods. Stat. Sci. 30, 443–467 (2015)MathSciNetCrossRefGoogle Scholar
  25. Marin, J.M., Mengersen, K., Robert, C.P.: Bayesian modelling and inference on mixtures of distributions. In: Dey, D.K., Rao, C.R. (eds.) Essential Bayesian Models, pp. 253–300. North-Holland, Amsterdam (2011)Google Scholar
  26. Møller, J., Pettitt, A., Reeves, R., Berthelsen, K.: An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika 93, 451–458 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. Murray, I., Ghahramani, Z., MacKay, D.: Mcmc for doubly-intractable distributions. In: Proceedings of the Twenty-Second Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI-06), AUAI Press, Arlington, Virginia, pp. 359–366 (2006)Google Scholar
  28. Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Stuct. Algorithms 9, 223–252 (1996)MathSciNetCrossRefMATHGoogle Scholar
  29. Reeves, R., Pettitt, A.N.: Efficient recursions for general factorisable models. Biometrika 91, 751–757 (2004)MathSciNetCrossRefMATHGoogle Scholar
  30. Riggan, W.B., Creason, J.P., Nelson, W.C., Manton, K.G., Woodbury, M.A., Stallard, E., Pellom, A.C., Beaubier, J.: U.S. Cancer Mortality Rates and Trends, 1950–1979, vol. IV (U.S. Goverment Printing Office, Washington, DC: Maps, U.S. Environmental Protection Agency) (1987)Google Scholar
  31. Sherman, M., Apanasovich, T.V., Carroll, R.J.: On estimation in binary autologistic spatial models. J. Stat. Comput. Simul. 76, 167–179 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. Tjelmeland, H., Austad, H.: Exact and approximate recursive calculations for binary Markov random fields defined on graphs. J. Comput. Graphical Stat. 21, 758–780 (2012)MathSciNetCrossRefGoogle Scholar
  33. Viterbi, A.J.: Error bounds for convolutional codes and an asymptotic optimum decoding algorithm. IEEE Trans. Inf. Theory 13, 260–269 (1967)CrossRefMATHGoogle Scholar
  34. Walker, S.: Posterior sampling when the normalising constant is unknown. Commun. Stat. Simul. Comput. 40, 784–792 (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.If P& COsloNorway

Personalised recommendations