Information-theoretic limits of Bayesian network structure learning

Asish Ghoshal, Jean Honorio
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:767-775, 2017.

Abstract

In this paper, we study the information-theoretic limits of learning the structure of Bayesian networks (BNs), on discrete as well as continuous random variables, from a finite number of samples. We show that the minimum number of samples required by any procedure to recover the correct structure grows as $Ω(m)$ and $Ω(k \log m + (k^2)/m)$ for non-sparse and sparse BNs respectively, where m is the number of variables and k is the maximum number of parents per node. We provide a simple recipe, based on an extension of the Fano’s inequality, to obtain information-theoretic limits of structure recovery for any exponential family BN. We instantiate our result for specific conditional distributions in the exponential family to characterize the fundamental limits of learning various commonly used BNs, such as conditional probability table based networks, Gaussian BNs, noisy-OR networks, and logistic regression networks. En route to obtaining our main results, we obtain tight bounds on the number of sparse and non-sparse essential-DAGs. Finally, as a byproduct, we recover the information-theoretic limits of sparse variable selection for logistic regression.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-ghoshal17a, title = {{Information-theoretic limits of Bayesian network structure learning}}, author = {Ghoshal, Asish and Honorio, Jean}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {767--775}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/ghoshal17a/ghoshal17a.pdf}, url = {https://proceedings.mlr.press/v54/ghoshal17a.html}, abstract = {In this paper, we study the information-theoretic limits of learning the structure of Bayesian networks (BNs), on discrete as well as continuous random variables, from a finite number of samples. We show that the minimum number of samples required by any procedure to recover the correct structure grows as $Ω(m)$ and $Ω(k \log m + (k^2)/m)$ for non-sparse and sparse BNs respectively, where m is the number of variables and k is the maximum number of parents per node. We provide a simple recipe, based on an extension of the Fano’s inequality, to obtain information-theoretic limits of structure recovery for any exponential family BN. We instantiate our result for specific conditional distributions in the exponential family to characterize the fundamental limits of learning various commonly used BNs, such as conditional probability table based networks, Gaussian BNs, noisy-OR networks, and logistic regression networks. En route to obtaining our main results, we obtain tight bounds on the number of sparse and non-sparse essential-DAGs. Finally, as a byproduct, we recover the information-theoretic limits of sparse variable selection for logistic regression.} }
Endnote
%0 Conference Paper %T Information-theoretic limits of Bayesian network structure learning %A Asish Ghoshal %A Jean Honorio %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-ghoshal17a %I PMLR %P 767--775 %U https://proceedings.mlr.press/v54/ghoshal17a.html %V 54 %X In this paper, we study the information-theoretic limits of learning the structure of Bayesian networks (BNs), on discrete as well as continuous random variables, from a finite number of samples. We show that the minimum number of samples required by any procedure to recover the correct structure grows as $Ω(m)$ and $Ω(k \log m + (k^2)/m)$ for non-sparse and sparse BNs respectively, where m is the number of variables and k is the maximum number of parents per node. We provide a simple recipe, based on an extension of the Fano’s inequality, to obtain information-theoretic limits of structure recovery for any exponential family BN. We instantiate our result for specific conditional distributions in the exponential family to characterize the fundamental limits of learning various commonly used BNs, such as conditional probability table based networks, Gaussian BNs, noisy-OR networks, and logistic regression networks. En route to obtaining our main results, we obtain tight bounds on the number of sparse and non-sparse essential-DAGs. Finally, as a byproduct, we recover the information-theoretic limits of sparse variable selection for logistic regression.
APA
Ghoshal, A. & Honorio, J.. (2017). Information-theoretic limits of Bayesian network structure learning. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:767-775 Available from https://proceedings.mlr.press/v54/ghoshal17a.html.

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