Geometry, moments and conditional independence trees with hidden variables

Geometry, moments and conditional independence trees with hidden variables

Raffaella Settimi and Jim Q. Smith

Source: Ann. Statist. Volume 28, Number 4 (2000), 1179-1205.

Abstract

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.

Primary Subjects: 62F15

Secondary Subjects: 62H17, 68R10

Keywords: Conditional independence; Bayesian networks; Bayesian multinomial models; model identifiability

Full-text: Open access

PDF File (241 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1015956712
Mathematical Reviews number (MathSciNet): MR1811324
Digital Object Identifier: doi:10.1214/aos/1015956712
Zentralblatt MATH identifier: 01828979

back to Table of Contents

References

Char, B., Geddes, K., Gonnet, G., Leong, B. and Monogan, M. (1995). MAPLE V Library Reference Manual. Springer, New York.

Chvat´al, V. (1983). Linear Programming. Freeman, New York.

Mathematical Reviews (MathSciNet): MR86g:90062

Cowell, R. G. (1998). Mixture reduction via predictive scores. Statist. Comput. 8 97-103.

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.

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

De Leeuw, J., van der Heijden, P. G. M. and Verboon, P. (1990). A latent time-budget model. Statist. Neerlandica 44 1-22.

Mathematical Reviews (MathSciNet): MR91e:62155

Feller, W. (1971). An Introduction to Probability Theory and its Applications 2. Wiley, New York.

Mathematical Reviews (MathSciNet): MR42:5292

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.

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.

Mathematical Reviews (MathSciNet): MR1617218

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.

Geisser, S. (1993). Predictive Inference: An Introduction. Chapman and Hall, London.

Mathematical Reviews (MathSciNet): MR95k:62006

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.

Mathematical Reviews (MathSciNet): MR82a:62070

Hartshorne, R. (1977). Algebraic Geometry. Springer, New York.

Mathematical Reviews (MathSciNet): MR57:3116

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

Mathematical Reviews (MathSciNet): MR98g:62001

Madigan, D. and York, J. (1995). Bayesian graphical models for discrete data. Internat. Statist. Rev. 63 215-232.

Zentralblatt MATH: 0834.62003

McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.

Mathematical Reviews (MathSciNet): MR88k:62004

Pistone, G., Riccomagno, E. and Wynn, H. P. (1999). Gr¨obner bases and factorisation in discrete probability and Bayes. Statist. Comput. (Special issue for the Workshop on Symbolic Computation, CRM, Montreal.)

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.

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.

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.

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.

Mathematical Reviews (MathSciNet): MR96m:68152

Spiegelhalter, D. J., Dawid, A. P. Lauritzen, S. L. and Cowell, R. G. (1993). Bayesian analysis of expert systems. Statist. Sci. 8 219-282.

Mathematical Reviews (MathSciNet): MR94j:62011

Spirtes, P., Richardson, T. and Meek, C. (1997). The dimensionality of mixed ancestral graphs. Technical Report CMU-PHIL-83, Dept. Philosophy, Carnegie Mellon Univ.

Streitberg, B. (1990). Lancaster interactions revisited. Ann. Statist. 18 1878-1885.

Mathematical Reviews (MathSciNet): MR92c:62008

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.

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.

Mathematical Reviews (MathSciNet): MR88h:62050

previous :: next