Previous |  Up |  Next


Bayesian networks; causal Markov condition; information theory; information inequalities; common ancestors; causal inference
Given a fixed dependency graph $G$ that describes a Bayesian network of binary variables $X_1, \dots, X_n$, our main result is a tight bound on the mutual information $I_c(Y_1, \dots, Y_k) = \sum_{j=1}^k H(Y_j)/c - H(Y_1, \dots, Y_k)$ of an observed subset $Y_1, \dots, Y_k$ of the variables $X_1, \dots, X_n$. Our bound depends on certain quantities that can be computed from the connective structure of the nodes in $G$. Thus it allows to discriminate between different dependency graphs for a probability distribution, as we show from numerical experiments.
[1] Allman, E. S., Rhodes, J. A.: Reconstructing Evolution: New Mathematical and Computational Advances, chapter Phylogenetic invariants. Oxford University Press, 2007. MR 2307988
[2] Ay, N.: A refinement of the common cause principle. Discrete Appl. Math. 157 (2009), 10, 2439-2457. DOI 10.1016/j.dam.2008.06.032 | MR 2527961
[3] Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2001. MR 1864966 | Zbl 1220.05117
[4] Cover, T. M., Thomas, J. A.: Elements of Information Theory. Second edition. Wiley, 2006. MR 2239987
[5] Friedman, N.: Inferring cellular networks using probabilistic graphical models. Science 303 (2004), 5659, 799-805. DOI 10.1126/science.1094068
[6] Peters, J., Mooij, J., Janzing, D., Schölkopf, B.: Causal discovery with continuous additive noise models. arXiv 1309.6779 (2013).
[7] Lauritzen, S. L.: Graphical Models. Oxford Science Publications, Clarendon Press, 1996. MR 1419991
[8] Lauritzen, S. L., Sheehan, N. A.: Graphical models for genetic analyses. Statist. Sci. 18 (2003), 489-514. DOI 10.1214/ss/1081443232 | MR 2059327 | Zbl 1055.62126
[9] Pearl, J.: Causality: Models, Reasoning and Inference. Cambridge University Press, 2000. MR 1744773 | Zbl 1188.68291
[10] Reichenbach, H., Reichenbach, M.: The Direction of Time. California Library Reprint Series, University of California Press, 1956.
[11] Socolar, J. E. S., Kauffman, S. A.: Scaling in ordered and critical random boolean networks. Phys. Rev. Lett. 90 (2003), 068702. DOI 10.1103/PhysRevLett.90.068702
[12] Steudel, B., Ay, N.: Information-theoretic inference of common ancestors. CoRR, abs/1010.5720, 2010.
[13] Studený, M.: Probabilistic Conditional Independence Structures. Information Science and Statistics. Springer, 2005. Zbl 1070.62001
Partner of
EuDML logo