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A graphical characterization of lattice conditional independence models

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Abstract

Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of non-monotone missing data patterns and of nonnested dependent linear regression models (≡ seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graph-theoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (possibly superexponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities.

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References

  1. T.W. Andersson, Maximum likelihood estimates for a multivariate normal distribution when some observations are missing, J. Amer. Statist. Assoc. 52 (1957) 200–203.

    Article  MathSciNet  Google Scholar 

  2. S.A. Andersson, The lattice structure of orthogonal linear models and orthogonal variance component models, Scand. J. Statist. 17 (1990) 287–319.

    MATH  MathSciNet  Google Scholar 

  3. S.A. Andersson, D. Madigan and M.D. Perlman, On the Markov equivalence of chain graphs, undirected graphs, and acyclic digraphs, Scand. J. Statist. 24 (1997) 81–102.

    Article  MATH  MathSciNet  Google Scholar 

  4. S.A. Andersson, D. Madigan and M.D. Perlman, A characterization of Markov equivalence classes for acyclic digraphs, Ann. Statist. (1997) (in press).

  5. S.A. Andersson, D. Madigan, M.D. Perlman and C.M. Triggs, On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs, J. Statist. Planning and Inference 48 (1995) 25–46.

    Article  MATH  MathSciNet  Google Scholar 

  6. S.A. Andersson and M.D. Perlman, Lattice models for conditional independence in a multivariate normal distribution. Preprint (Institute of Mathematical Statistics, University of Copenhagen and Department of Statistics, University of Washington, 1988).

  7. S.A. Andersson and M.D. Perlman, Lattice-ordered conditional independence models for missing data, Statist. Probability Letters 12 (1991) 465–486.

    Article  MATH  MathSciNet  Google Scholar 

  8. S.A. Andersson and M.D. Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993) 1318–1358.

    MATH  MathSciNet  Google Scholar 

  9. S.A. Andersson and M.D. Perlman, Normal linear models with lattice conditional independence restrictions, in: Multivariate Analysis and Its Applications, eds. T.W. Anderson, K.-T. Fang and I. Olkin, IMS Lecture Notes-Monograph Series, Vol. 24 (1994) pp. 97–110.

  10. S.A. Andersson and M.D. Perlman, Testing lattice conditional independence models, J. Multivariate Analysis 53 (1995) 18–38.

    Article  MATH  MathSciNet  Google Scholar 

  11. S.A. Andersson and M.D. Perlman, Unbiasedness of the likelihood ratio test for lattice conditional independence models, J. Multivariate Analysis 53 (1995) 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  12. S.A. Andersson and M.D. Perlman, Normal linear regression models with recursive graphical Markov structure, Submitted to J. Multivariate Analysis (1997).

  13. J.R.S. Blair and B. Peyton, An introduction to chordal graphs and clique trees, in: Graph Theory and Sparse Matrix Computation, eds. A. George, J.R. Gilbert and J.W.H. Liu (Springer-Verlag, New York, 1993) pp. 1–29.

    Google Scholar 

  14. D.M. Chickering, A transformational characterization of equivalent Bayesian network structures, in: Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference, eds. P. Besnard and S. Hanks (Morgan Kaufmann, San Francisco, 1995) pp. 87–98.

    Google Scholar 

  15. D.R. Cox and N. Wermuth, Multivariate Dependencies: Models, Analysis, and Interpretation (Chapman and Hall, London, 1996).

    Google Scholar 

  16. B.A. Davey and H.A. Priestley, Introduction to Lattices and Order (Cambridge U. Press, Cambridge, UK, 1990).

    Google Scholar 

  17. A.P. Dawid and S.L. Lauritzen, Hyper Markov laws in the statistical analysis of decomposable graphical models, Ann. Statist. 21 (1993) 1272–1317.

    MATH  MathSciNet  Google Scholar 

  18. D. Edwards, Introduction to Graphical Modeling (Springer, New York, 1995).

    Google Scholar 

  19. M. Frydenberg, The chain graph Markov property, Scand. J. Statist. 17 (1990) 333–353.

    MATH  MathSciNet  Google Scholar 

  20. F.V. Jensen, An Introduction to Bayesian Network (U. College London Press, London, 1996).

    Google Scholar 

  21. J.T.A. Koster, Markov properties of nonrecursive causal models, Ann. Statist. 24 (1996) 2148–2177.

    Article  MATH  MathSciNet  Google Scholar 

  22. S.L. Lauritzen, Graphical Models (Oxford University Press, Oxford, 1996).

    Google Scholar 

  23. S.L. Lauritzen, A.P. Dawid, B.N. Larsen and H.-G. Leimer, Independence properties of directed Markov fields, Networks 20 (1990) 491–505.

    MATH  MathSciNet  Google Scholar 

  24. S.L. Lauritzen and D.J. Spiegelhalter, Local computations with probabilities on graphical structures and their application to expert systems (with discussion), J. Royal Statist. Soc., Series B 50 (1988) 157–224.

    MATH  MathSciNet  Google Scholar 

  25. D. Madigan and A.E. Raftery, Model selection and accounting for model uncertainty in graphical models using Occam's window, J. Amer. Statist. Assoc. 89 (1994) 1535–1546.

    Article  MATH  Google Scholar 

  26. C. Meek, Causal inference and causal explanation with background knowledge, in: Uncertainty in Artificial Intelligence: Proceedings of the Eleventh Conference, eds. P. Besnard and S. Hanks (Morgan Kaufmann, San Francisco, 1995) pp. 403–410.

    Google Scholar 

  27. R.E. Neapolitan, Probabilistic Reasoning in Expert Systems (Wiley, New York, 1990).

    Google Scholar 

  28. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, San Mateo, CA, 1988).

    Google Scholar 

  29. M.D. Perlman and L. Wu, Lattice conditional independence models for missing data in contingency tables (1996) (in preparation).

  30. B.W. Peyton, A. Pothen and X. Yuan, Partitioning a chordal graph into transitive subgraphs for parallel sparse triangular solution, Linear Algebra Appl. 192 (1993) 329–354.

    Article  MATH  MathSciNet  Google Scholar 

  31. D.J. Spiegelhalter, A.P. Dawid, S.L. Lauritzen and R.G. Cowell, Bayesian analysis in expert systems (with discussion), Statist. Science 8 (1993) 219–283.

    MATH  MathSciNet  Google Scholar 

  32. D.J. Spiegelhalter and S.L. Lauritzen, Sequential updating of conditional probabilities on directed graphical structures, Networks 20 (1990) 579–605.

    MATH  MathSciNet  Google Scholar 

  33. P. Spirtes, C. Glymour and R. Scheines, Causation, Prediction, and Search (Springer, New York, 1993).

    Google Scholar 

  34. T. Verma and J. Pearl, Equivalence and synthesis of causal models, in: Uncertainty in Artificial Intelligence: Proceedings of the Sixth Conference, eds. M. Henrion et al. (North-Holland, 1990) pp. 220–227.

  35. T. Verma and J. Pearl, An algorithm for deciding if a set of observed independencies has a causal explanation, in: Uncertainty in Artificial Intelligence: Proceedings of the Eighth Conference, eds. D. Dubois et al. (Morgan Kaufmann, San Francisco, 1992) pp. 323–330.

    Google Scholar 

  36. N. Wermuth and N. Lauritzen, Graphical and recursive models for contingency tables, Biometrika 70 (1983) 537–552.

    Article  MATH  MathSciNet  Google Scholar 

  37. J.L. Whittaker, Graphical Models in Applied Multivariate Statistics (Wiley, New York, 1990).

    Google Scholar 

  38. S. Wright, Correlation and causation, J. Agricultural Research 20 (1921) 557–585.

    Google Scholar 

  39. J. York, D. Madigan, I. Heuch and R.T. Lie, Estimating a proportion of birth defects by double sampling: a Bayesian approach incorporating covariates and model uncertainty, Applied Statistics 44 (1995) 227–242.

    Article  MATH  Google Scholar 

  40. A. Zellner, An efficient method of estimating seemingly unrelated regression equations and tests for aggregation bias, J. Amer. Statist. Assoc. 57 (1962) 348–368.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Indiana University, USA

    Steen A. Andersson, David Madigan, Michael D. Perlman & Christopher M. Triggs

  2. University of Washington, USA

    Steen A. Andersson, David Madigan, Michael D. Perlman & Christopher M. Triggs

  3. University of Auckland, USA

    Steen A. Andersson, David Madigan, Michael D. Perlman & Christopher M. Triggs

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  1. Steen A. Andersson
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  2. David Madigan
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  3. Michael D. Perlman
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  4. Christopher M. Triggs
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Andersson, S.A., Madigan, D., Perlman, M.D. et al. A graphical characterization of lattice conditional independence models. Annals of Mathematics and Artificial Intelligence 21, 27–50 (1997). https://doi.org/10.1023/A:1018901032102

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