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Multiway Dependence in Exponential Family Conditional Distributions

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  • Lee, Jaehyung
  • Kaiser, Mark S.
  • Cressie, Noel

Abstract

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the "pairwise-only dependence" assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192-225) formulated exponential family "auto-models" and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:79:y:2001:i:2:p:171-190
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    References listed on IDEAS

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    1. 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.
    2. Kaiser, Mark S. & Cressie, Noel, 1997. "Modeling Poisson variables with positive spatial dependence," Statistics & Probability Letters, Elsevier, vol. 35(4), pages 423-432, November.
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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. 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.
    3. Dreassi, Emanuela & Rigo, Pietro, 2017. "A note on compatibility of conditional autoregressive models," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 9-16.
    4. Cécile Hardouin & Noel Cressie, 2018. "Two-scale spatial models for binary data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(1), pages 1-24, March.

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