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The Construction of Multivariate Distributions from Markov Random Fields

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  • Kaiser, Mark S.
  • Cressie, Noel

Abstract

We address the problem of constructing and identifying a valid joint probability density function from a set of specified conditional densities. The approach taken is based on the development of relations between the joint and the conditional densities using Markov random fields (MRFs). We give a necessary and sufficient condition on the support sets of the random variables to allow these relations to be developed. This condition, which we call the Markov random field support condition, supercedes a common assumption known generally as the positivity condition. We show how these relations may be used in reverse order to construct a valid model from specification of conditional densities alone. The constructive process and the role of conditions needed for its application are illustrated with several examples, including MRFs with multiway dependence and a spatial beta process.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:73:y:2000:i:2:p:199-220
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    References listed on IDEAS

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    1. Cuadras, C. M., 1992. "Probability distributions with given multivariate marginals and given dependence structure," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 51-66, July.
    2. Li, Haijun & Scarsini, Marco & Shaked, Moshe, 1996. "Linkages: A Tool for the Construction of Multivariate Distributions with Given Nonoverlapping Multivariate Marginals," Journal of Multivariate Analysis, Elsevier, vol. 56(1), pages 20-41, January.
    3. Cramer, Erhard, 1998. "Conditional Iterative Proportional Fitting for Gaussian Distributions," Journal of Multivariate Analysis, Elsevier, vol. 65(2), pages 261-276, May.
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    Cited by:

    1. Emily Casleton & Daniel J. Nordman & Mark S. Kaiser, 2022. "Modeling Transitivity in Local Structure Graph Models," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 389-417, June.
    2. Christopher K. Wikle, 2003. "Hierarchical Models in Environmental Science," International Statistical Review, International Statistical Institute, vol. 71(2), pages 181-199, August.
    3. R. Reeves, 2004. "Efficient recursions for general factorisable models," Biometrika, Biometrika Trust, vol. 91(3), pages 751-757, September.
    4. Wang, Yuchung J. & Kuo, Kun-Lin, 2010. "Compatibility of discrete conditional distributions with structural zeros," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 191-199, January.
    5. 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.
    6. Kopciuszewski, Pawel, 2004. "An extension of the factorization theorem to the non-positive case," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 118-130, January.
    7. 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.
    8. Mark S. Kaiser & Petruţa C. Caragea, 2009. "Exploring Dependence with Data on Spatial Lattices," Biometrics, The International Biometric Society, vol. 65(3), pages 857-865, September.
    9. Berti, Patrizia & Dreassi, Emanuela & Rigo, Pietro, 2014. "Compatibility results for conditional distributions," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 190-203.
    10. Nail Kashaev & Natalia Lazzati, 2019. "Peer Effects in Random Consideration Sets," Papers 1904.06742, arXiv.org, revised May 2021.
    11. Dreassi, Emanuela & Rigo, Pietro, 2017. "A note on compatibility of conditional autoregressive models," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 9-16.
    12. 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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