The Annals of Statistics

Marginalization and conditioning for LWF chain graphs

Kayvan Sadeghi

Full-text: Open access


In this paper, we deal with the problem of marginalization over and conditioning on two disjoint subsets of the node set of chain graphs (CGs) with the LWF Markov property. For this purpose, we define the class of chain mixed graphs (CMGs) with three types of edges and, for this class, provide a separation criterion under which the class of CMGs is stable under marginalization and conditioning and contains the class of LWF CGs as its subclass. We provide a method for generating such graphs after marginalization and conditioning for a given CMG or a given LWF CG. We then define and study the class of anterial graphs, which is also stable under marginalization and conditioning and contains LWF CGs, but has a simpler structure than CMGs.

Article information

Ann. Statist., Volume 44, Number 4 (2016), 1792-1816.

Received: May 2014
Revised: January 2016
First available in Project Euclid: 7 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section
Secondary: 62A99: None of the above, but in this section

$c$-separation criterion chain graph independence model LWF Markov property $m$-separation marginalization and conditioning mixed graph


Sadeghi, Kayvan. Marginalization and conditioning for LWF chain graphs. Ann. Statist. 44 (2016), no. 4, 1792--1816. doi:10.1214/16-AOS1451.

Export citation


  • [1] Andersson, S. A., Madigan, D. and Perlman, M. D. (2001). Alternative Markov properties for chain graphs. Scand. J. Statist. 28 33–85.
  • [2] Cox, D. R. and Wermuth, N. (1993). Linear dependencies represented by chain graphs. Statist. Sci. 8 204–218, 247–283. With comments and a rejoinder by the authors.
  • [3] Drton, M. (2009). Discrete chain graph models. Bernoulli 15 736–753.
  • [4] Evans, R. J. and Richardson, T. S. (2014). Markovian acyclic directed mixed graphs for discrete data. Ann. Statist. 42 1452–1482.
  • [5] Frydenberg, M. (1990). The chain graph Markov property. Scand. J. Statist. 17 333–353.
  • [6] Geiger, D., Heckerman, D., King, H. and Meek, C. (2001). Stratified exponential families: Graphical models and model selection. Ann. Statist. 29 505–529.
  • [7] Kiiveri, H., Speed, T. P. and Carlin, J. B. (1984). Recursive causal models. J. Austral. Math. Soc. Ser. A 36 30–52.
  • [8] Koster, J. T. A. (2002). Marginalizing and conditioning in graphical models. Bernoulli 8 817–840.
  • [9] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. The Clarendon Press, Oxford.
  • [10] Lauritzen, S. L. and Spiegelhalter, D. J. (1988). Local computations with probabilities on graphical structures and their application to expert systems. J. Roy. Statist. Soc. Ser. B 50 157–224.
  • [11] Lauritzen, S. L. and Wermuth, N. (1989). Graphical models for associations between variables, some of which are qualitative and some quantitative. Ann. Statist. 17 31–57.
  • [12] Marchetti, G. M. and Lupparelli, M. (2011). Chain graph models of multivariate regression type for categorical data. Bernoulli 17 827–844.
  • [13] Pearl, J. (2009). Causality: Models, Reasoning, and Inference, 2nd ed. Cambridge Univ. Press, New York.
  • [14] Peña, J. M. (2009). Faithfulness in chain graphs: The discrete case. Internat. J. Approx. Reason. 50 1306–1313.
  • [15] Peña, J. M. (2011). Faithfulness in chain graphs: The Gaussian case. In Proceedings of the 14th International Conference on Artificial Intelligence and Statistics (AISTATS 2011) 15 588–599.
  • [16] Peña, J. M. (2014). Marginal AMP chain graphs. Internat. J. Approx. Reason. 55 1185–1206.
  • [17] Richardson, T. (2003). Markov properties for acyclic directed mixed graphs. Scand. J. Stat. 30 145–157.
  • [18] Richardson, T. and Spirtes, P. (2002). Ancestral graph Markov models. Ann. Statist. 30 962–1030.
  • [19] Sadeghi, K. (2013). Stable mixed graphs. Bernoulli 19 2330–2358.
  • [20] Sadeghi, K. (2016). Supplement to “Marginalization and conditioning for LWF chain graphs”. DOI:10.1214/16-AOS1451SUPP.
  • [21] Shpitser, I. and Pearl, J. (2008). Dormant independence. In Proceedings of the Twenty-Third AAAI Conference on Artificial Inteligence 1081–1087. AAAI Press, Menlo Park.
  • [22] Studeny, M. (1998). Bayesian networks from the point of view of chain graphs. In UAI 496–503. Morgan Kaufmann, San Francisco, CA.
  • [23] Studený, M. (2005). Probabilistic Conditional Independence Structures. Springer, London.
  • [24] Studený, M. and Bouckaert, R. R. (1998). On chain graph models for description of conditional independence structures. Ann. Statist. 26 1434–1495.
  • [25] Verma, T. and Pearl, J. (1990). Equivalence and synthesis of causal models. In Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence (UAI-90) 220–227. Elsevier, New York.
  • [26] Wermuth, N. (2011). Probability distributions with summary graph structure. Bernoulli 17 845–879.
  • [27] Wermuth, N. and Sadeghi, K. (2012). Sequences of regressions and their independences. TEST 21 215–252.
  • [28] Wermuth, N., Wiedenbeck, M. and Cox, D. R. (2006). Partial inversion for linear systems and partial closure of independence graphs. BIT 46 883–901.

Supplemental materials