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Markov Properties for Acyclic Directed Mixed Graphs

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  • THOMAS RICHARDSON

Abstract

We consider acyclic directed mixed graphs, in which directed edges (x→y) and bi‐directed edges (x↔y) may occur. A simple extension of Pearl's d‐separation criterion, called m‐separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m‐separation criterion for arbitrary distributions.

Suggested Citation

  • Thomas Richardson, 2003. "Markov Properties for Acyclic Directed Mixed Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 145-157, March.
    • Handle: RePEc:bla:scjsta:v:30:y:2003:i:1:p:145-157
    DOI: 10.1111/1467-9469.00323
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    Cited by:

    1. Boitani, Andrea & Punzo, Chiara, 2019. "Banks’ leverage behaviour in a two-agent new Keynesian model," Journal of Economic Behavior & Organization, Elsevier, vol. 162(C), pages 347-359.
    2. Colombi, R. & Giordano, S., 2012. "Graphical models for multivariate Markov chains," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 90-103.
    3. Christian Gische & Manuel C. Voelkle, 2022. "Beyond the Mean: A Flexible Framework for Studying Causal Effects Using Linear Models," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 868-901, September.
    4. Robin J. Evans & Thomas S. Richardson, 2013. "Marginal log-linear parameters for graphical Markov models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 743-768, September.
    5. Robin J. Evans, 2016. "Graphs for Margins of Bayesian Networks," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 625-648, September.
    6. Eichler, M. & Didelez, V., 2009. "On Granger-causality and the effect of interventions in time series," Research Memorandum 003, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    7. Monia Lupparelli & Giovanni M. Marchetti & Wicher P. Bergsma, 2009. "Parameterizations and Fitting of Bi‐directed Graph Models to Categorical Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 559-576, September.
    8. Claudia Tarantola & Ioannis Ntzoufras, 2012. "Bayesian Analysis of Graphical Models of Marginal Independence for Three Way Contingency Tables," Quaderni di Dipartimento 172, University of Pavia, Department of Economics and Quantitative Methods.
    9. Lorenza Rossi & Emilio Zanetti Chini, 2016. "Firms’ Dynamics and Business Cycle: New Disaggregated Data," DEM Working Papers Series 123, University of Pavia, Department of Economics and Management.
    10. 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.
    11. Ntzoufras, Ioannis & Tarantola, Claudia, 2013. "Conjugate and conditional conjugate Bayesian analysis of discrete graphical models of marginal independence," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 161-177.
    12. Alberto Roverato, 2015. "Log-mean Linear Parameterization for Discrete Graphical Models of Marginal Independence and the Analysis of Dichotomizations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 627-648, June.
    13. Colombi, R. & Giordano, S., 2015. "Multiple hidden Markov models for categorical time series," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 19-30.
    14. Ioannis Ntzoufras & Claudia Tarantola & Monia Lupparelli, 2018. "Probability Based Independence Sampler for Bayesian Quantitative Learning in Graphical Log-Linear Marginal Models," DEM Working Papers Series 149, University of Pavia, Department of Economics and Management.

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