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Alternative Markov Properties for Chain Graphs

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  • Steen A. Andersson
  • David Madigan
  • Michael D. Perlman

Abstract

Graphical Markov models use graphs to represent possible dependences among statistical variables. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs (CG): graphs that can be used to represent both structural and associative dependences simultaneously and that include both undirected graphs (UG) and acyclic directed graphs (ADG) as special cases. Here an alternative Markov property (AMP) for CGs is introduced and shown to be the Markov property satisfied by a block‐recursive linear system with multivariate normal errors. This model can be decomposed into a collection of conditional normal models, each of which combines the features of multivariate linear regression models and covariance selection models, facilitating the estimation of its parameters. In the general case, necessary and sufficient conditions are given for the equivalence of the LWF and AMP Markov properties of a CG, for the AMP Markov equivalence of two CGs, for the AMP Markov equivalence of a CG to some ADG or decomposable UG, and for other equivalences. For CGs, in some ways the AMP property is a more direct extension of the ADG Markov property than is the LWF property.

Suggested Citation

  • Steen A. Andersson & David Madigan & Michael D. Perlman, 2001. "Alternative Markov Properties for Chain Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 33-85, March.
    • Handle: RePEc:bla:scjsta:v:28:y:2001:i:1:p:33-85
    DOI: 10.1111/1467-9469.00224
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    Citations

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    Cited by:

    1. Castelletti, Federico & Peluso, Stefano, 2021. "Equivalence class selection of categorical graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 164(C).
    2. Marchetti, Giovanni M., 2006. "Independencies Induced from a Graphical Markov Model After Marginalization and Conditioning: The R Package ggm," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 15(i06).
    3. repec:jss:jstsof:15:i06 is not listed on IDEAS
    4. Johnson, Devin S. & Hoeting, Jennifer A., 2011. "Properties of graphical regression models for multidimensional categorical data," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1471-1475, October.
    5. Colombi, R. & Giordano, S., 2012. "Graphical models for multivariate Markov chains," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 90-103.
    6. Yang Ni & Veerabhadran Baladandayuthapani & Marina Vannucci & Francesco C. Stingo, 2022. "Bayesian graphical models for modern biological applications," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(2), pages 197-225, June.
    7. Andersson, Steen A. & Klein, Thomas, 2010. "On Riesz and Wishart distributions associated with decomposable undirected graphs," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 789-810, April.
    8. Federico Castelletti, 2020. "Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures," International Statistical Review, International Statistical Institute, vol. 88(3), pages 752-775, December.
    9. Roberto Colombi & Sabrina Giordano, 2013. "Monotone dependence in graphical models for multivariate Markov chains," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(7), pages 873-885, October.
    10. Nanny Wermuth & Kayvan Sadeghi, 2012. "Sequences of regressions and their independences," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 215-252, June.
    11. Vassilios Bazinas & Bent Nielsen, 2022. "Causal Transmission in Reduced-Form Models," Econometrics, MDPI, vol. 10(2), pages 1-25, March.
    12. Federica Nicolussi & Fulvia Mecatti, 2016. "A smooth subclass of graphical models for chain graph: towards measuring gender gaps," Quality & Quantity: International Journal of Methodology, Springer, vol. 50(1), pages 27-41, January.
    13. S. A. Andersson & G. G. Wojnar, 2004. "Wishart Distributions on Homogeneous Cones," Journal of Theoretical Probability, Springer, vol. 17(4), pages 781-818, October.

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