An asymptotic behaviour of the marginal likelihood for general markov models

Journal of Machine Learning Research

Volumn 12, Issue , 2011, Pages 3283-3310

  Zwiernik, Piotr ^a  

^a Institute for Pure and Applied Mathematics   (United States)

Author keywords

Bayesian networks; BIC; Marginal likelihood; Real logcanonical threshold; Singular models; Tree models

Indexed keywords

ASYMPTOTIC BEHAVIOUR; BAYESIAN INFORMATION CRITERION; BIC; GRAPHICAL MODEL; GROWTH BEHAVIOR; HIDDEN VARIABLE; MARGINAL LIKELIHOOD; MARKOV MODEL; NAIVE BAYES MODELS; REAL LOGCANONICAL THRESHOLD; REGULARITY CONDITION; TREE MODELS;

ASYMPTOTIC ANALYSIS; BAYESIAN NETWORKS; FORESTRY;

HIDDEN MARKOV MODELS;

BINARY SYSTEMS; GRAPHIC METHODS; MATHEMATICAL MODELS; VARIABLES;

EID: 84855406025     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Article

References (20)

1. Vladimir I. Arnold, Sabir M. Guseǐn-Zade, and Aleksandr N. Varchenko. Singularities of Differentiable Maps, volume II. Birkhäuser, 1988.
2. Edward Bierstone and Pierre D. Milman. Semianalytic and subanalytic sets. Publications Mathématiques de l'IHÉS, 67(1):5-42, 1988.
3. David Maxwell Chickering and David Heckerman. Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables. Machine Learning, 29(2):181-212, 1997. (Pubitemid 127510038)
4. Gregory F. Cooper and Edward Herskovits. A Bayesian method for the induction of probabilistic networks from data. Machine Learning, 9(4):309-347, 1992.
5. Ewgenij Gawrilow and Michael Joswig. Geometric reasoning with polymake. arXiv:math.CO/ 0507273, 2005.
6. Mark Goresky and Robert MacPherson. Stratified Morse Theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. ISBN 3-540-17300-5.
7. Dominique Haughton. On the choice of a model to fit data from an exponential family. Ann. Statist, 16(1):342-355, 1988.
8. David Heckerman, Dan Geiger, and David Maxwell Chickering. Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning, 20(3):197-243, 1995.
9. Robert Lazarsfeld. Positivity in Algebraic Geometry. A Series of Modern Surveys in Mathematics. Springer Verlag, 2004.
10. Shaowei Lin. Asymptotic Approximation of Marginal Likelihood Integrals. arXiv:1003.5338, November 2011. submitted.
11. RaduMihaescu and Lior Pachter. Combinatorics of least-squares trees. Proceedings of the National Academy of Sciences of the United States of America, 105(36):13206, 2008.
12. Judea Pearl and Michael Tarsi. Structuring causal trees. J. Complexity, 2(1):60-77, 1986. ISSN 0885-064X. Complexity of approximately solved problems (Morningside Heights, N.Y., 1985).
13. Dmitry Rusakov and Dan Geiger. Asymptotic model selection for naive Bayesian networks. J. Mach. Learn. Res., 6:1-35 (electronic), 2005. ISSN 1532-4435.
14. Morihiko Saito. On real log canonical thresholds. arXiv:0707.2308, 2007.
15. Gideon Schwarz. Estimating the dimension of a model. Annals of Statistics, 6(2):461-464, 1978.
16. Charles Semple andMike Steel. Phylogenetics, volume 24 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2003. ISBN 0-19-850942-1.
17. Sumio Watanabe. Algebraic Geometry and Statistical Learning Theory. Number 25 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2009. ISBN-13: 9780521864671.
18. Keisuke Yamazaki and Sumio Watanabe. Newton diagram and stochastic complexity in mixture of binomial distributions. In Algorithmic Learning Theory, volume 3244 of Lecture Notes in Comput. Sci., pages 350-364. Springer, Berlin, 2004. (Pubitemid 41050303)
19. Piotr Zwiernik and Jim Q. Smith. Implicit inequality constraints in a binary tree model. Electron. J. Statist., 5:1276-1312, 2011a. ISSN 1935-7524. doi: 10.1214/11-EJS640.
20. Piotr Zwiernik and Jim Q. Smith. Tree-cumulants and the geometry of binary tree models. to appear in Bernoulli, 2011b.