On the Sample Complexity of Learning Bayesian Networks

In recent years there has been an increasing interest in learning Bayesian networks from data. One of the most effective methods for learning such networks is based on the minimum description length (MDL) principle. Previous work has shown that this learning procedure is asymptotically successful: with probability one, it will converge to the target distribution, given a sufficient number of samples. However, the rate of this convergence has been hitherto unknown. In this work we examine the sample complexity of MDL based learning procedures for Bayesian networks. We show that the number of samples needed to learn an e-close approximation (in terms of entropy distance) with confidence δ is O ((1/e)4/3 log 1/e log 1/δ log log 1/δ). This means that the sample complexity is a low-order polynomial in the error threshold and sub-linear in the confidence bound. We also discuss how the constants in this term depend on the complexity of the target distribution. Finally, we address questions of asymptotic minimality and propose a method for using the sample complexity results to speed up the learning process.

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  H. Akaike.  A new look at the statistical model identification , 1974 .

[3]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[4]  Rodney W. Johnson,et al.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy , 1980, IEEE Trans. Inf. Theory.

[5]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[6]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[7]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[8]  Andrew R. Barron,et al.  Minimum complexity density estimation , 1991, IEEE Trans. Inf. Theory.

[9]  Joe Suzuki,et al.  A Construction of Bayesian Networks from Databases Based on an MDL Principle , 1993, UAI.

[10]  Stuart J. Russell,et al.  Decision Theoretic Subsampling for Induction on Large Databases , 1993, ICML.

[11]  Klaus-Uwe Höffgen,et al.  Learning and robust learning of product distributions , 1993, COLT '93.

[12]  Wai Lam,et al.  LEARNING BAYESIAN BELIEF NETWORKS: AN APPROACH BASED ON THE MDL PRINCIPLE , 1994, Comput. Intell..

[13]  Nir Friedman,et al.  Learning Bayesian Networks with Local Structure , 1996, UAI.

[14]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[15]  Jorma Rissanen,et al.  Stochastic Complexity in Statistical Inquiry , 1989, World Scientific Series in Computer Science.