Learning Bayesian networks is NP-hard

In this paper, we provide new complexity results for algorithms that learn discretevariable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest model able to represent the generative distribution exactly. Our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset. We show that identifying high-scoring structures is NP-hard, even when we are given an independence oracle, an inference oracle, and/or an information oracle. Our negative results also apply when learning discrete-variable Bayesian networks in which each node has at most k parents, for all k ≥ 3.