The max-min hill-climbing Bayesian network structure learning algorithm

Ioannis Tsamardinos, Laura E. Brown, Constantin F. Aliferis

Research output: Contribution to journalArticlepeer-review

896 Scopus citations


We present a new algorithm for Bayesian network structure learning, called Max-Min Hill-Climbing (MMHC). The algorithm combines ideas from local learning, constraint-based, and search-and-score techniques in a principled and effective way. It first reconstructs the skeleton of a Bayesian network and then performs a Bayesian-scoring greedy hill-climbing search to orient the edges. In our extensive empirical evaluation MMHC outperforms on average and in terms of various metrics several prototypical and state-of-the-art algorithms, namely the PC, Sparse Candidate, Three Phase Dependency Analysis, Optimal Reinsertion, Greedy Equivalence Search, and Greedy Search. These are the first empirical results simultaneously comparing most of the major Bayesian network algorithms against each other. MMHC offers certain theoretical advantages, specifically over the Sparse Candidate algorithm, corroborated by our experiments. MMHC and detailed results of our study are publicly available at

Original languageEnglish (US)
Pages (from-to)31-78
Number of pages48
JournalMachine Learning
Issue number1
StatePublished - Oct 2006

Bibliographical note

Funding Information:
Acknowledgments The first and third author are supported by the National Library of Medicine grant LM-7948-01. The second author is supported by the National Library of Medicine grant T15 LM07450-01. We are in debt to Alexander Statnikov for parts of the code, testing, and useful suggestions and to Yerbolat Dosbayev for help with some implementation issues. We would like to thank Sonia M. Leach and Douglas H. Fisher for their useful feedback. We would like to thank the authors of the Sparse Candidate, Tetrad, and Optimal Reinsertion for making publicly available their algorithms and systems for inclusion in this study. Finally, we would like to thank the editor and the anonymous reviewers for their helpful feedback and for providing a draft of the computational complexity comparison between the Greedy Search and MMHC.


  • Bayesian networks
  • Graphical models
  • Structure learning

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