We present a new approach for learning the structure of a treewidth-bounded Bayesian Network (BN). The key to our approach is applying an exact method (based on MaxSAT) locally, to improve the score of a heuristically computed BN. This approach allows us to scale the power of exact methods -- so far only applicable to BNs with several dozens of random variables -- to large BNs with several thousands of random variables. Our experiments show that our method improves the score of BNs provided by state-of-the-art heuristic methods, often significantly.
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