6,492 research outputs found
Exact Learning of Bounded Tree-width Bayesian Networks
Abstract Inference in Bayesian networks is known to be NP-hard, but if the network has bounded treewidth, 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 fixedparameter tractable in the tree-width of the super-structure
Efficient learning of Bayesian networks with bounded tree-width
Learning Bayesian networks with bounded tree-width has attracted much attention recently, because low tree-width allows exact inference to be performed efficiently. Some existing methods [24,29] tackle the problem by using k-trees to learn the optimal Bayesian network with tree-width up to k. Finding the best k-tree, however, is computationally intractable. In this paper, we propose a sampling method to efficiently find representative k-trees by introducing an informative score function to characterize the quality of a k-tree. To further improve the quality of the k-trees, we propose a probabilistic hill climbing approach that locally refines the sampled k-trees. The proposed algorithm can efficiently learn a quality Bayesian network with tree-width at most k. Experimental results demonstrate that our approach is more computationally efficient than the exact methods with comparable accuracy, and outperforms most existing approximate methods
Advances in Learning Bayesian Networks of Bounded Treewidth
This work presents novel algorithms for learning Bayesian network structures
with bounded treewidth. Both exact and approximate methods are developed. The
exact method combines mixed-integer linear programming formulations for
structure learning and treewidth computation. The approximate method consists
in uniformly sampling -trees (maximal graphs of treewidth ), and
subsequently selecting, exactly or approximately, the best structure whose
moral graph is a subgraph of that -tree. Some properties of these methods
are discussed and proven. The approaches are empirically compared to each other
and to a state-of-the-art method for learning bounded treewidth structures on a
collection of public data sets with up to 100 variables. The experiments show
that our exact algorithm outperforms the state of the art, and that the
approximate approach is fairly accurate.Comment: 23 pages, 2 figures, 3 table
Learning Bounded Treewidth Bayesian Networks with Thousands of Variables
We present a method for learning treewidth-bounded Bayesian networks from
data sets containing thousands of variables. Bounding the treewidth of a
Bayesian greatly reduces the complexity of inferences. Yet, being a global
property of the graph, it considerably increases the difficulty of the learning
process. We propose a novel algorithm for this task, able to scale to large
domains and large treewidths. Our novel approach consistently outperforms the
state of the art on data sets with up to ten thousand variables
On the Relationship between Sum-Product Networks and Bayesian Networks
In this paper, we establish some theoretical connections between Sum-Product
Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be
converted into a BN in linear time and space in terms of the network size. The
key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent
the local conditional probability distributions at each node in the resulting
BN by exploiting context-specific independence (CSI). The generated BN has a
simple directed bipartite graphical structure. We show that by applying the
Variable Elimination algorithm (VE) to the generated BN with ADD
representations, we can recover the original SPN where the SPN can be viewed as
a history record or caching of the VE inference process. To help state the
proof clearly, we introduce the notion of {\em normal} SPN and present a
theoretical analysis of the consistency and decomposability properties. We
conclude the paper with some discussion of the implications of the proof and
establish a connection between the depth of an SPN and a lower bound of the
tree-width of its corresponding BN.Comment: Full version of the same paper to appear at ICML-201
Cutset Sampling for Bayesian Networks
The paper presents a new sampling methodology for Bayesian networks that
samples only a subset of variables and applies exact inference to the rest.
Cutset sampling is a network structure-exploiting application of the
Rao-Blackwellisation principle to sampling in Bayesian networks. It improves
convergence by exploiting memory-based inference algorithms. It can also be
viewed as an anytime approximation of the exact cutset-conditioning algorithm
developed by Pearl. Cutset sampling can be implemented efficiently when the
sampled variables constitute a loop-cutset of the Bayesian network and, more
generally, when the induced width of the networks graph conditioned on the
observed sampled variables is bounded by a constant w. We demonstrate
empirically the benefit of this scheme on a range of benchmarks
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