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On the Characterization of a Finite Random Field by Conditional Distribution and its Gibbs Form

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  • Linda Khachatryan

    (National Academy of Science of the Republic of Armenia)

  • Boris S. Nahapetian

    (National Academy of Science of the Republic of Armenia)

Abstract

In this paper, we show that the methods of mathematical statistical physics can be successfully applied to random fields in finite volumes. As a result, we obtain simple necessary and sufficient conditions for the existence and uniqueness of a finite random field with a given system of one-point conditional distributions. Using the axiomatic (without the notion of potential) definition of Hamiltonian, we show that any finite random field is Gibbsian. We also apply the proposed approach to Markov random fields.

Suggested Citation

  • Linda Khachatryan & Boris S. Nahapetian, 2023. "On the Characterization of a Finite Random Field by Conditional Distribution and its Gibbs Form," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1743-1761, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01209-6
    DOI: 10.1007/s10959-022-01209-6
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    References listed on IDEAS

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    1. Berti, Patrizia & Dreassi, Emanuela & Rigo, Pietro, 2014. "Compatibility results for conditional distributions," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 190-203.
    2. Chen, Hua Yun, 2010. "Compatibility of conditionally specified models," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 670-677, April.
    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.
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