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19
Efficient structure learning of Bayesian networks using constraints
 Journal of Machine Learning Research
"... This paper addresses the problem of learning Bayesian network structures from data based on score functions that are decomposable. It describes properties that strongly reduce the time and memory costs of many known methods without losing global optimality guarantees. These properties are derived fo ..."
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Cited by 30 (7 self)
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This paper addresses the problem of learning Bayesian network structures from data based on score functions that are decomposable. It describes properties that strongly reduce the time and memory costs of many known methods without losing global optimality guarantees. These properties are derived for different score criteria such as Minimum Description Length (or Bayesian Information Criterion), Akaike Information Criterion and Bayesian Dirichlet Criterion. Then a branchandbound algorithm is presented that integrates structural constraints with data in a way to guarantee global optimality. As an example, structural constraints are used to map the problem of structure learning in Dynamic Bayesian networks into a corresponding augmented Bayesian network. Finally, we show empirically the benefits of using the properties with stateoftheart methods and with the new algorithm, which is able to handle larger data sets than before.
Exact structure discovery in Bayesian networks with less space
 In Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence (UAI
, 2009
"... The fastest known exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space 2 n n O(1). The usage of these algorithms is limited to networks on at most around 25 nodes mainly due to the space requirement. Here, we study space–time tradeoffs for finding ..."
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Cited by 29 (7 self)
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The fastest known exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space 2 n n O(1). The usage of these algorithms is limited to networks on at most around 25 nodes mainly due to the space requirement. Here, we study space–time tradeoffs for finding an optimal network structure. When little space is available, we apply the Gurevich– Shelah recurrence—originally proposed for the Hamiltonian path problem—and obtain time 2 2n−s n O(1) in space 2 s n O(1) for any s = n/2,n/4,n/8,...; we assume the indegree of each node is bounded by a constant. For the more practical setting with moderate amounts of space, we present a novel scheme. It yields running time 2 n (3/2) p n O(1) in space 2 n (3/4) p n O(1) for any p = 0,1,...,n/2; these bounds hold as long as the indegrees are at most 0.238n. Furthermore, the latter scheme allows easy and efficient parallelization beyond previous algorithms. We also explore empirically the potential of the presented techniques. 1
Finding Optimal Bayesian Network Given a SuperStructure
"... Classical approaches used to learn Bayesian network structure from data have disadvantages in terms of complexity and lower accuracy of their results. However, a recent empirical study has shown that a hybrid algorithm improves sensitively accuracy and speed: it learns a skeleton with an independenc ..."
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Cited by 17 (0 self)
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Classical approaches used to learn Bayesian network structure from data have disadvantages in terms of complexity and lower accuracy of their results. However, a recent empirical study has shown that a hybrid algorithm improves sensitively accuracy and speed: it learns a skeleton with an independency test (IT) approach and constrains on the directed acyclic graphs (DAG) considered during the searchandscore phase. Subsequently, we theorize the structural constraint by introducing the concept of superstructure S, which is an undirected graph that restricts the search to networks whose skeleton is a subgraph of S. We develop a superstructure constrained optimal search (COS): its time complexity is upper bounded by O(γm n), where γm < 2 depends on the maximal degree m of S. Empirically, complexity depends on the average degree ˜m and sparse structures allow larger graphs to be calculated. Our algorithm is faster than an optimal search by several orders and even finds more accurate results when given a sound superstructure. Practically, S can be approximated by IT approaches; significance level of the tests controls its sparseness, enabling to control the tradeoff between speed and accuracy. For incomplete superstructures, a greedily postprocessed version (COS+) still enables to significantly outperform other heuristic searches. Keywords: subset Bayesian networks, structure learning, optimal search, superstructure, connected 1.
Learning Optimal Bounded Treewidth Bayesian Networks via Maximum Satisfiability
, 2014
"... Bayesian network structure learning is the wellknown computationally hard problem of finding a directed acyclic graph structure that optimally describes given data. A learned structure can then be used for probabilistic inference. While exact inference in Bayesian networks is in general NPhard, ..."
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Cited by 11 (5 self)
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Bayesian network structure learning is the wellknown computationally hard problem of finding a directed acyclic graph structure that optimally describes given data. A learned structure can then be used for probabilistic inference. While exact inference in Bayesian networks is in general NPhard, it is tractable in networks with low treewidth. This provides good motivations for developing algorithms for the NPhard problem of learning optimal bounded treewidth Bayesian networks (BTWBNSL). In this work, we develop a novel scorebased approach to BTWBNSL, based on casting BTWBNSL as weighted partial Maximum satisfiability. We demonstrate empirically that the approach scales notably better than a recent exact dynamic programming algorithm for BTWBNSL.
A space–time tradeoff for permutation problems
 In Proceedings of the ACMSIAM Symposium on Discrete Algorithms (SODA
, 2010
"... Many combinatorial problems—such as the traveling salesman, feedback arcset, cutwidth, and treewidth problem— can be formulated as finding a feasible permutation of n elements. Typically, such problems can be solved by dynamic programming in time and space O ∗ (2 n), by divide and conquer in time O ..."
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Cited by 8 (4 self)
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Many combinatorial problems—such as the traveling salesman, feedback arcset, cutwidth, and treewidth problem— can be formulated as finding a feasible permutation of n elements. Typically, such problems can be solved by dynamic programming in time and space O ∗ (2 n), by divide and conquer in time O ∗ (4 n) and polynomial space, or by a combination of the two in time O ∗ (4 n 2 −s) and space O ∗ (2 s) for s = n, n/2, n/4,.... Here, we show that one can improve the tradeoff to time O ∗ (T n) and space O ∗ (S n) with T S < 4 at any √ 2 < S < 2. The idea is to find a small family of “thin ” partial orders on the n elements such that every linear order is an extension of one member of the family. Our construction is optimal within a natural class of partial order families. 1
Utilizing evolutionary information and gene expression data for estimating gene networks with Bayesian network models
, 2005
"... Since microarray gene expression data do not contain sufficient information for estimating accurate gene networks, other biological information has been considered to improve the estimated networks. Recent studies have revealed that highly conserved proteins that exhibit similar expression patterns ..."
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Cited by 5 (0 self)
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Since microarray gene expression data do not contain sufficient information for estimating accurate gene networks, other biological information has been considered to improve the estimated networks. Recent studies have revealed that highly conserved proteins that exhibit similar expression patterns in different organisms, have almost the same function in each organism. Such conserved proteins are also known to play similar roles in terms of the regulation of genes. Therefore, this evolutionary information can be used to refine regulatory relationships among genes, which are estimated from gene expression data. We propose a statistical method for estimating gene networks from gene expression data by utilizing evolutionarily conserved relationships between genes. Our method simultaneously estimates two gene networks of two distinct organisms, with a Bayesian network model utilizing the evolutionary information so that gene expression data of one organism helps to estimate the gene network of the other. We show the effectiveness of the method through the analysis on Saccharomyces cerevisiae and Homo sapiens cell cycle gene expression data. Our method was successful in estimating gene networks that capture many known relationships as well as several unknown relationships which are likely to be novel. Supplementary information is available at
Methods to Accelerate the Learning of Bayesian Network Structures
 PROCEEDINGS OF THE 2007 UK WORKSHOP ON COMPUTATIONAL INTELLIGENCE
, 2007
"... Bayesian networks have become a standard technique in the representation of uncertain knowledge. This paper proposes methods that can accelerate the learning of a Bayesian network structure from a data set. These methods are applicable when learning an equivalence class of Bayesian network structure ..."
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Cited by 3 (0 self)
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Bayesian networks have become a standard technique in the representation of uncertain knowledge. This paper proposes methods that can accelerate the learning of a Bayesian network structure from a data set. These methods are applicable when learning an equivalence class of Bayesian network structures whilst using a score and search strategy. They work by constraining the number of validity tests that need to be done and by caching the results of validity tests. The results of experiments show that the methods improve the performance of algorithms that search through the space of equivalence classes multiple times and that operate on wide data sets. The experiments were performed by sampling data from six standard Bayesian networks and running an ant colony optimization algorithm designed to learn a Bayesian network equivalence class. 1
Finding Optimal Bayesian Networks Using Precedence Constraints
, 2013
"... 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 ..."
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Cited by 2 (0 self)
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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 2 n, 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 2 n. 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.
Bayesian structure discovery in Bayesian networks with less space
"... Current exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space within a polynomial factor of 2 n. For practical use, the space requirement is the bottleneck, which motivates trading space against time. Here, previous results on finding an optimal n ..."
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Current exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space within a polynomial factor of 2 n. For practical use, the space requirement is the bottleneck, which motivates trading space against time. Here, previous results on finding an optimal network structure in less space are extended in two directions. First, we consider the problem of computing the posterior probability of a given arc set. Second, we operate with the general partial order framework and its specialization to bucket orders, introduced recently for related permutation problems. The main technical contribution is the development of a fast algorithm for a novel zeta transform variant, which may be of independent interest. 1
Increasing Feasibility of Optimal Gene Network Estimation
 Genome Informatics
, 2004
"... Disentangling networks of regulation of gene expression is a major challenge in the field of computational biology. Harvesting the information contained in microarray data sets is a promising approach towards this challenge. We propose an algorithm for the optimal estimation of Bayesian networks fro ..."
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Disentangling networks of regulation of gene expression is a major challenge in the field of computational biology. Harvesting the information contained in microarray data sets is a promising approach towards this challenge. We propose an algorithm for the optimal estimation of Bayesian networks from microarray data, which reduces the CPU time and memory consumption of previous algorithms. We prove that the space complexity can be reduced from O(n )toO(2 ), and that the expected calculation time can be reduced from O(n )toO(n ), where n is the number of genes. We make intrinsic use of a limitation of the maximal number of regulators of each gene, which has biological as well as statistical justifications. The improvements are significant for some applications in research.