Article
Full entry |
PDF
(1.7 MB)
Feedback
PDF
(1.7 MB)
Feedback
Keywords:
factorizability; directed graph
factorizability; directed graph
Summary:
Four notions of factorizability over arbitrary directed graphs are examined. For acyclic graphs they coincide and are identical with the usual factorization of probability distributions in Markov models. Relations between the factorizations over circuits are described in detail including nontrivial counterexamples. Restrictions on the cardinality of state spaces cause that a factorizability with respect to some special cyclic graphs implies the factorizability with respect to their, more simple, strict edge-subgraphs. This gives sometimes the possibility to break circuits and get back to the acyclic, well-understood case.
References:
[1] Berman A., Plemmons R. J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York – San Francisco – London 1979 MR 0544666 | Zbl 0815.15016
[2] Cox D. R., Wermuth N.: Multivariate Dependencies. (Monographs on Statistics and Applied Probability 67.) Chapman & Hall, London 1996 MR 1456990 | Zbl 0880.62124
[3] Dawid A. P.: Conditional independence for statistical operations. Ann. Statist. 8 (1980), 598–617 DOI 10.1214/aos/1176345011 | MR 0568723 | Zbl 0434.62006
[4] Lauritzen S. L.: Graphical Models. Clarendon Press, Oxford 1996 MR 1419991 | Zbl 1055.62126
[5] Koster J. T. A.: Gibbs and Markov properties of graphs. Ann. Math. and Artificial Inteligence 21 (1997), 13–26 DOI 10.1023/A:1018948915264 | MR 1479006 | Zbl 0895.68115
[6] Koster J. T. A.: Markov properties of non–recursive causal models. Ann. Statist. 24 (1996), 2148–2177 DOI 10.1214/aos/1069362315 | MR 1421166
[7] Spirtes P.: Directed cyclic graphical representation of feedback models. In: Proceedings of the Eleventh Conference on Uncertainty and Artificial Inteligence (P. Besnard and S. Hanks, eds.), Morgan Kaufman Publ. Inc., San Mateo 1995
[8] Strohmeier B.: Cyclical Causal Networks and Knowledge Integration. Ph.D. Dissertation. Fakultät für Wirtschaftswissenschaften, Universität Bielefeld 1996
[9] Vajda I.: Theory of Statistical Inference and Information. Kluwer Academic Publishers, Dordrecht – Boston – London 1989 Zbl 0711.62002
[10] Whittaker J.: Graphical Models in Applied Multivariate Statistics. J. Wiley, New York 1990 MR 1112133 | Zbl 1151.62053
Search
Partner of
