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Bayesian reference analysis for Gaussian Markov random fields

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  • Ferreira, Marco A.R.
  • De Oliveira, Victor
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    Abstract

    Gaussian Markov random fields (GMRF) are important families of distributions for the modeling of spatial data and have been extensively used in different areas of spatial statistics such as disease mapping, image analysis and remote sensing. GMRFs have been used for the modeling of spatial data, both as models for the sampling distribution of the observed data and as models for the prior of latent processes/random effects; we consider mainly the former use of GMRFs. We study a large class of GMRF models that includes several models previously proposed in the literature. An objective Bayesian analysis is presented for the parameters of the above class of GMRFs, where explicit expressions for the Jeffreys (two versions) and reference priors are derived, and for each of these priors results on posterior propriety of the model parameters are established. We describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters, and study frequentist properties of the Bayesian inferences resulting from the use of these automatic priors. Finally, we illustrate the use of the proposed GMRF model and reference prior for studying the spatial variability of lip cancer cases in the districts of Scotland over the period 1975-1980.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 98 (2007)
    Issue (Month): 4 (April)
    Pages: 789-812

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    Handle: RePEc:eee:jmvana:v:98:y:2007:i:4:p:789-812

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    Keywords: Frequentist properties Integrated likelihood Jeffreys prior Objective Bayesian analysis Reference prior Spatial statistics;

    References

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    1. Berger J.O. & De Oliveira V. & Sanso B., 2001. "Objective Bayesian Analysis of Spatially Correlated Data," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1361-1374, December.
    2. Y. Yang, 1995. "Invariance of the reference prior under reparametrization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 4(1), pages 83-94, June.
    3. H�vard Rue, 2001. "Fast sampling of Gaussian Markov random fields," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 325-338.
    4. Gamerman, Dani & Moreira, Ajax R. B. & Rue, Havard, 2003. "Space-varying regression models: specifications and simulation," Computational Statistics & Data Analysis, Elsevier, vol. 42(3), pages 513-533, March.
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    Cited by:
    1. Victor De Oliveira & Marco Ferreira, 2011. "Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields," Metrika, Springer, vol. 74(2), pages 167-183, September.
    2. Vinicius Mayrink & Dani Gamerman, 2009. "On computational aspects of Bayesian spatial models: influence of the neighboring structure in the efficiency of MCMC algorithms," Computational Statistics, Springer, vol. 24(4), pages 641-669, December.

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