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Binary Markov Mesh Models and Symmetric Markov Random Fields: Some Results on their Equivalence

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  • Noel Cressie

    (The Ohio State University)

  • Craig Liu

    (Iowa State University)

Abstract

In this article, we focus on statistical models for binary data on a regular two-dimensional lattice. We study two classes of models, the Markov mesh models (MMMs) based on causal-like, asymmetric spatial dependence, and symmetric Markov random fields (SMFs) based on noncausal-like, symmetric spatial dependence. Building on results of Enting (1977), we give sufficient conditions for the asymmetrically defined binary MMMs (of third order) to be equivalent to a symmetrically defined binary SMF. Although not every binary SMF can be written as a binary MMM, our results show that many can. For such SMFs, their joint distribution can be written in closed form and their realizations can be simulated with just one pass through the lattice. An important consequence of the latter observation is that there are nontrivial spatial processes for which exact probabilities can be used to benchmark the performance of Markov-chain-Monte-Carlo and other algorithms.

Suggested Citation

  • Noel Cressie & Craig Liu, 2001. "Binary Markov Mesh Models and Symmetric Markov Random Fields: Some Results on their Equivalence," Methodology and Computing in Applied Probability, Springer, vol. 3(1), pages 5-34, March.
    • Handle: RePEc:spr:metcap:v:3:y:2001:i:1:d:10.1023_a:1011461923517
    DOI: 10.1023/A:1011461923517
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    References listed on IDEAS

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    1. Cressie, Noel & Davidson, Jennifer L., 1998. "Image analysis with partially ordered markov models," Computational Statistics & Data Analysis, Elsevier, vol. 29(1), pages 1-26, November.
    2. J. Møller, 1999. "Perfect simulation of conditionally specified models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 251-264.
    3. Kaiser, Mark S. & Cressie, Noel, 2000. "The Construction of Multivariate Distributions from Markov Random Fields," Journal of Multivariate Analysis, Elsevier, vol. 73(2), pages 199-220, May.
    4. Lee, Jaehyung & Kaiser, Mark S. & Cressie, Noel, 2001. "Multiway Dependence in Exponential Family Conditional Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(2), pages 171-190, November.
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    Cited by:

    1. Emilio De Santis & Mauro Piccioni, 2008. "Exact Simulation for Discrete Time Spin Systems and Unilateral Fields," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 105-120, March.

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