  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 epsilon-close approximation (in terms of entropy distance)
with confidence delta is O((1/epsilon)^(4/3)log(1/epsilon)log(1/delta)loglog
(1/delta)). 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.

Machine Learning;Machine Learning