Conference Paper

# Learning Optimal Bayesian Networks Using A* Search.

DOI: 10.5591/978-1-57735-516-8/IJCAI11-364 Conference: IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011

Source: DBLP

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**ABSTRACT:**We consider the problem of finding a directed acyclic graph (DAG) that optimizes a decomposable Bayesian network score. While in a favorable case an optimal DAG can be found in polynomial time, in the worst case the fastest known algorithms rely on dynamic programming across the node subsets, taking time and space 2n, to within a factor polynomial in the number of nodes n. In practice, these algorithms are feasible to networks of at most around 30 nodes, mainly due to the large space requirement. Here, we generalize the dynamic programming approach to enhance its feasibility in three dimensions: first, the user may trade space against time; second, the proposed algorithms easily and efficiently parallelize onto thousands of processors; third, the algorithms can exploit any prior knowledge about the precedence relation on the nodes. Underlying all these results is the key observation that, given a partial order P on the nodes, an optimal DAG compatible with P can be found in time and space roughly proportional to the number of ideals of P, which can be significantly less than 2n. Considering sufficiently many carefully chosen partial orders guarantees that a globally optimal DAG will be found. Aside from the generic scheme, we present and analyze concrete tradeoff schemes based on parallel bucket orders.Journal of Machine Learning Research 01/2013; 14(1):1387-1415. · 3.42 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this work, we empirically evaluate the capability of various scoring functions of Bayesian networks for recovering true underlying structures. Similar investigations have been carried out before, but they typically relied on approximate learning algorithms to learn the network structures. The suboptimal structures found by the approximation methods have unknown quality and may affect the reliability of their conclusions. Our study uses an optimal algorithm to learn Bayesian network structures from datasets generated from a set of gold standard Bayesian networks. Because all optimal algorithms always learn equivalent networks, this ensures that only the choice of scoring function affects the learned networks. Another shortcoming of the previous studies stems from their use of random synthetic networks as test cases. There is no guarantee that these networks reflect real-world data. We use real-world data to generate our gold-standard structures, so our experimental design more closely approximates real-world situations. A major finding of our study suggests that, in contrast to results reported by several prior works, the Minimum Description Length (MDL) (or equivalently, Bayesian information criterion (BIC)) consistently outperforms other scoring functions such as Akaike's information criterion (AIC), Bayesian Dirichlet equivalence score (BDeu), and factorized normalized maximum likelihood (fNML) in recovering the underlying Bayesian network structures. We believe this finding is a result of using both datasets generated from real-world applications rather than from random processes used in previous studies and learning algorithms to select high-scoring structures rather than selecting random models. Other findings of our study support existing work, e.g., large sample sizes result in learning structures closer to the true underlying structure; the BDeu score is sensitive to the parameter settings; and the fNML performs pretty well on small datasets. We also tested a greedy hill climbing algorithm and observed similar results as the optimal algorithm.BMC Bioinformatics 09/2012; 13 Suppl 15:S14. · 3.02 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) super-structure. The super-structure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed super-structure as a natural generalization of its undirected counterpart. Our results apply to several variants of score-based Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth, and in linear time if in addition the super-structure has bounded maximum degree. Furthermore, we show that if the directed super-structure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Similarly, exact Bayesian network structure learning remains NP-hard for "almost acyclic" directed super-structures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).Journal of Artificial Intelligence Research 02/2014; 46(1). · 1.06 Impact Factor

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