Zwiernik, Piotr (2010) Asymptotic model selection and identifiability of directed tree models with hidden variables. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).

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Abstract

The standard Bayesian Information Criterion (BIC) is derived under some regularity conditions which are not always satisfied by the graphical models with hidden variables. In this paper we derive the BIC score for Bayesian networks in the case when the data is binary and the underlying graph is a rooted tree and all the inner nodes represent hidden variables. This provides a direct generalization of a similar formula given by Rusakov and Geiger in [10]. Geometric results obtained in this paper are complementary to the results in the previous paper [18] extending our understanding of this class of models. The main tool used in this paper is the connection between asymptotic approximation of Laplace integrals and the real log-canonical threshold.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Bayesian statistical decision theory, Trees (Graph theory)
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	2010
Volume:	Vol.2010
Number:	No.6
Number of Pages:	27
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] V. Arnold, S. Gusein-Zade, and A. Varchenko, Singularities of differentiable maps, vol. II, Birkhauser, 1988. [2] E. Gawrilow and M. Joswig, Geometric reasoning with polymake. arXiv:math.CO/0507273, 2005. [3] D. Geiger, D. Heckerman, H. King, and C. Meek, Stratified exponential families: graph- ical models and model selection, Ann. Statist., 29 (2001), pp. 505{529. [4] D. Haughton, On the choice of a model to fit data from an exponential family, Ann. Statist, 16 (1988), pp. 342{355. [5] J. Kollar, Singularities of pairs, in Proceedings of Symposia in Pure Mathematics, vol. 62, 1997, pp. 221{288. [6] R. Lazarsfeld, Positivity in algebraic geometry, A Series of Modern Surveys in Mathematics, Springer Verlag, 2004. [7] S. Lin, Asymptotic approximation of marginal likelihood integrals. in preparation, 2010. [8] R. Mihaescu and L. Pachter, Combinatorics of least-squares trees, Proceedings of the National Academy of Sciences of the United States of America, 105 (2008), p. 13206. [9] V. Moulton and M. Steel, Peeling phylogenetic 'oranges', Advances in Applied Mathematics, 33 (2004), pp. 710{727. [10] D. Rusakov and D. Geiger, Asymptotic model selection for naive Bayesian networks, J. Mach. Learn. Res., 6 (2005), pp. 1{35 (electronic). [11] M. Saito, On real log canonical thresholds, Arxiv preprint arXiv:0707.2308 v, 1 (2007). [12] G. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978), pp. 461{ 464. [13] C. Semple and M. Steel, Phylogenetics, vol. 24 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2003. [14] R. P. Stanley, Enumerative combinatorics. Volume I, no. 49 in Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2002. [15] S. Watanabe, Algebraic Analysis for Nonidentifiable Learning Machines, Neural Computation, 13 (2001), pp. 899{933. [16] S. Watanabe, Algebraic Geometry and Statistical Learning Theory, no. 25 in Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2009. ISBN-13: 9780521864671. [17] K. Yamazaki and S. Watanabe, Newton diagram and stochastic complexity in mixture of binomial distributions, in Algorithmic learning theory, vol. 3244 of Lecture Notes in Comput. Sci., Springer, Berlin, 2004, pp. 350{364. [18] P. Zwiernik and J. Q. Smith, The geometry of conditional independence tree models with hidden variables. Arxiv preprint arXiv:0904.1980, January 2010.
URI:	http://wrap.warwick.ac.uk/id/eprint/35068

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