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# Exact Learning of Bounded Tree-width Bayesian Networks

*Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics*, PMLR 31:370-378, 2013.

#### Abstract

Inference in Bayesian networks is known to be NP-hard, but if the network has bounded tree-width, then inference becomes tractable. Not surprisingly, learning networks that closely match the given data and have a bounded tree-width has recently attracted some attention. In this paper we aim to lay groundwork for future research on the topic by studying the exact complexity of this problem. We give the first non-trivial exact algorithm for the NP-hard problem of finding an optimal Bayesian network of tree-width at most w, with running time 3^n n^w + O(1), and provide an implementation of this algorithm. Additionally, we propose a variant of Bayesian network learning with “super-structures”, and show that finding a Bayesian network consistent with a given super-structure is fixed-parameter tractable in the tree-width of the super-structure.