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Computational Bayesian Analysis of Hidden Markov Mesh Models
November 1997 (vol. 19 no. 11)
pp. 1296-1300

Abstract—Versions of the Gibbs Sampler are derived for the analysis of data from the hidden Markov Mesh random fields sometimes used in image analysis. This provides a numerical approach to the otherwise intractable Bayesian analysis of these problems. Detailed formulation is provided for particular examples based on Devijver's [4] Markov Mesh model, and the BUGS [20] package is used to do the computations. Theoretical aspects are discussed and a numerical study, based on image analysis, is reported.

[1] K. Abend,T.J. Hartley,, and L.N. Kanal,“Classification of binary random patterns,” IEEE Trans. Information Theory, vol. 11, no. 4, pp. 538-544, Oct. 1965.
[2] J. Besag, P.J. Green, D. Higdon, and K. Mengersen, "Bayesian Computation and Stochastic Systems (with Discussion)," Statistical Science, vol. 10, pp. 3-66, 1995.
[3] N.G. Best, M.K. Cowles, and S.K. Vines, CODA Manual version 0.30. MRC Biostatistics Unit, Cambridge, UK, 1995.
[4] P.A. Devijver, "Image Segmentation Using Causal Markov Random Field Models," Proc. Fourth Int'l Conf. Pattern Recognition, pp. 131-143, 1988.
[5] J. Diebolt and C.P. Robert, "Estimation of Finite Mixture Distributions Through Bayesian Sampling," J. Royal Statistical Soc. B, vol. 56, pp. 363-375, 1994.
[6] A.P. Dunmur and D.M. Titterington, "Parameter Estimation in Latent Structure Models," Tech. Report 96-2, Dept. of Statistics, Univ. of Glasgow, 1996.
[7] A.P. Dunmur and D.M. Titterington, "Computational Bayesian Analysis of Hidden Markov Mesh Models," Tech. Report 96-16, Dept. of Statistics, Univ. of Glasgow, 1996.
[8] A. Frigessi, P. di Stefano, C.-R. Hwang, and S.-J. Sheu, "Convergence Rates of the Gibbs Sampler, the Metropolis Algorithm and Other Single-Site Updating Dynamics," J. Royal Statistical Soc. B, vol. 55, pp. 205-220, 1993.
[9] A.E. Gelfand, S.E. Hills, A. Racine-Poon, and A.F.M. Smith, "Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling," J. Am. Statistical Assoc., vol. 85, pp. 972-985, 1990.
[10] A.E. Gelfand and A.F.M. Smith, "Sampling-Based Approaches to Calculating Marginal Densities," J. Am. Statistical Assoc., vol. 85, pp. 398-409, 1990.
[11] S. Geman and D. Geman, "Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, pp. 721-741, 1984.
[12] Practical Markov Chain Monte Carlo, W.R. Gilks, S.T. Richardson, and D.J. Spiegelhalter, eds. London: Chapman and Hall, 1996.
[13] A.J. Gray, J.W. Kay, and D.M. Titterington, "An Empirical Study of the Simulation of Various Models Used for Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, pp. 507-512, 1994.
[14] J. Heikkinen and H. Högmander, "Fully Bayesian Approach to Image Restoration with an Application in Biogeography," Applied Statistics, vol. 43, pp. 569-582, 1994.
[15] D.M. Higdon, V.E. Johnson, T.G. Turkington, J.E. Bowsher, D.R. Gilland, and R.J. Jaszczak, "Fully Bayesian Estimation of Gibbs Hyperparameters for Emission Computed Tomography Data" technical report, Inst. for Decision Sciences, Duke Univ., 1996.
[16] R.M. Neal, "Connectionist Learning of Belief Networks," Artificial Intelligence, vol. 56, pp. 71-113, 1992.
[17] W. Qian and D.M. Titterington, "Pixel Labeling for Three-Dimensional Scenes Based on Markov Mesh Models," Signal Processing, vol. 22, pp. 313-328, 1991.
[18] C.P. Robert, G. Celeux, and J. Diebolt, "Bayesian Estimation of Hidden Markov Chains: A Stochastic Implementation," Statistics&Probability Letters, vol. 16, pp. 77-83, 1993.
[19] T. Rydén and D.M. Titterington, "Computational Bayesian Analysis of Hidden Markov Models," J. Computational $ Graphical Statistics, to appear, 1998.
[20] D.J. Spiegelhalter, A. Thomas, N. Best, and W.R. Gilks, BUGS: Bayesian inference Using Gibbs Sampling, Version 0.50. MRC Biostatistics Unit, Cambridge, 1995.
[21] A. Thomas, D.J. Spiegelhalter, and W.R. Gilks, "BUGS: A Program to Perform Bayesian Inference Using Gibbs Sampling," Bayesian Statistics 4, J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith, eds., pp. 837-842.Oxford, UK: Clarendon Press, 1992.
[22] L. Tierney, "Markov Chains for Exploring Posterior Distributions," Annals of Statistics, vol. 22, no. 4, pp. 1,701-1,728, 1994.
[23] I.S. Weir, "Fully Bayesian Reconstructions from Single Photon Emission Computed Tomography Data," J. Am. Statistical Assoc., vol. 92, pp. 49-60, 1997.
[24] J. Whittaker, Graphical Models in Applied Multivariate Statistics. Wiley, 1990.

Index Terms:
Bayesian inference, Gibbs sampling, hidden Markov Mesh random field, Markov chain Monte Carlo.
Citation:
A.p. Dunmur, D.m. Titterington, "Computational Bayesian Analysis of Hidden Markov Mesh Models," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 11, pp. 1296-1300, Nov. 1997, doi:10.1109/34.632989
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