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Geometry, moments and conditional independence trees with hidden variables. (English) Zbl 1105.62321
Summary: We study the geometry of the parameter space for Bayesian directed graphical models with hidden variables that have a tree structure and where all the nodes are binary. We show that the conditional independence statements implicit in such models can be expressed in terms of polynomial relationships among the central moments. This algebraic structure will enable us to identify the inequality constraints on the space of the manifest variables that are induced by the conditional independence assumptions as well as determine the degree of unidentifiability of the parameters associated with the hidden variables. By understanding the geometry of the sample space under this class of models we shall propose and discuss simple diagnostic methods.

MSC:
62F15 Bayesian inference
05C90 Applications of graph theory
62H99 Multivariate analysis
Software:
Maple
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[1] Char, B., Geddes, K., Gonnet, G., Leong, B. and Monogan, M. (1995). MAPLE V Library Reference Manual. Springer, New York. · Zbl 0763.68046
[2] Chvatál, V. (1983). Linear Programming. Freeman, New York.
[3] Cowell, R. G. (1998). Mixture reduction via predictive scores. Statist. Comput. 8 97-103.
[4] Cox, D., Little, J. and O’Shea, D. (1991). Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, New York.
[5] Cox, D. R. and Wermuth, N. (1996). Multivariate Dependencies. Models Analysis and Interpretation. Chapman and Hall, London. · Zbl 0880.62124
[6] De Leeuw, J., van der Heijden, P. G. M. and Verboon, P. (1990). A latent time-budget model. Statist. Neerlandica 44 1-22. · Zbl 0718.62292
[7] Feller, W. (1971). An Introduction to Probability Theory and its Applications 2. Wiley, New York. · Zbl 0219.60003
[8] Geiger, D., Heckerman, D., King, H. and Meek, C. (1998). Stratified exponential families: graphical models and model selection. Technical Report MSR-TR-98-31, Microsoft Research Center, WA. · Zbl 1012.62012
[9] Geiger, D., Heckerman, D. and Meek, C. (1996). Asymptotic model selection for directed networks with hidden variables. In Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence 283-290. Morgan Kaufmann, San Mateo, CA. · Zbl 0910.68177
[10] Geiger, D. and Meek, C. (1998). Graphical models and exponential families. In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence 156-165. Morgan Kaufmann, San Mateo, CA.
[11] Geisser, S. (1993). Predictive Inference: An Introduction. Chapman and Hall, London. · Zbl 0824.62001
[12] Gilula, Z. (1979). Singular value decomposition of probability matrices: probabilistic aspects of latent dichotomous variables. Biometrika 66 339-344. Goodman, L. A. (1974a). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61 215-231. Goodman, L. A. (1974b). The analysis of systems of qualitative variables when some of the variables are unobservable. A modified latent structure approach. Amer. J. Sociology 79 1179-1259. JSTOR: · Zbl 0411.62003
[13] Hartshorne, R. (1977). Algebraic Geometry. Springer, New York. · Zbl 0367.14001
[14] Lauritzen, S. L. (1996). Graphical Models. Oxford Univ. Press. · Zbl 0907.62001
[15] Madigan, D. and York, J. (1995). Bayesian graphical models for discrete data. Internat. Statist. Rev. 63 215-232. · Zbl 0834.62003
[16] McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London. · Zbl 0732.62003
[17] Pistone, G., Riccomagno, E. and Wynn, H. P. (1999). Gröbner bases and factorisation in discrete probability and Bayes. Statist. Comput. (Special issue for the Workshop on Symbolic Computation, CRM, Montreal.)
[18] Ramoni, M. and Sebastiani, P. (1997). Learning Bayesian networks from incomplete databases. In Proceedings of the Thirteenth Conference on Uncertainty in Artificial Intelligence 401-408. Morgan Kaufmann, San Mateo, CA. · Zbl 1066.62011
[19] Settimi, R. and Smith, J. Q. (1998). On the geometry of Bayesian graphical models with hidden variables. In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence 472-479. Morgan Kaufman, San Mateo, CA.
[20] Settimi, R. and Smith, J. Q. (1999). Geometry, moments and Bayesian networks with hidden variables. In Proceedings of the Seventh International Workshop on Statistics and Artificial Intelligence 293-298. Morgan Kaufmann, San Mateo, CA. · Zbl 1105.62321
[21] Spiegelhalter, D. J. and Cowell, R. G. (1992). Learning in probabilistic expert systems. In Bayesian Statistics (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 4 447-466. Clarendon, Oxford.
[22] Spiegelhalter, D. J., Dawid, A. P. Lauritzen, S. L. and Cowell, R. G. (1993). Bayesian analysis of expert systems. Statist. Sci. 8 219-282. · Zbl 0955.62523
[23] Spirtes, P., Richardson, T. and Meek, C. (1997). The dimensionality of mixed ancestral graphs. Technical Report CMU-PHIL-83, Dept. Philosophy, Carnegie Mellon Univ.
[24] Streitberg, B. (1990). Lancaster interactions revisited. Ann. Statist. 18 1878-1885. · Zbl 0713.62056
[25] Swofford, D. L., Olsen, G. J., Waddell, P. J. and Hillis, D. M. (1996). Phylogenetic inference. In Molecular Systematics, 2nd ed. (Hillis, D. M., Moritz, C. and Mable, B. K., eds.) 407-514. Sinauer Assoc., Sunderland, MA.
[26] Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distribution by data augmentation (with discussion). J. Amer. Statist. Assoc. 82 528-550. JSTOR: · Zbl 0619.62029
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