## Large-Sample Learning of Bayesian Networks is NP-Hard

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David Maxwell Chickering, David Heckerman, Christopher Meek**; 5(Oct):1287--1330, 2004.

### Abstract

In this paper, we provide new complexity results for algorithms that learn discrete-variable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest structure for which the model is 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 any combination of one or more of the following hold: the generative distribution is perfect with respect to some DAG containing hidden variables; we are given an independence oracle; we are given an inference oracle; we are given an information oracle; we restrict potential solutions to structures in which each node has at most*k*parents, for all

*k*>=3.

Our proof relies on a new technical result that we establish in the appendices. In particular, we provide a method for constructing the local distributions in a Bayesian network such that the resulting joint distribution is provably perfect with respect to the structure of the network.

[abs][pdf]