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SIAM Journal on Algebraic and Discrete Methods
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Information Processing Letters
Approximating probabilistic inference in Bayesian belief networks is NP-hard
Artificial Intelligence
Graph classes: a survey
Learning Markov networks: maximum bounded tree-width graphs
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Finding a path is harder than finding a tree
Journal of Artificial Intelligence Research
UAI'99 Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence
Learning Factor Graphs in Polynomial Time and Sample Complexity
The Journal of Machine Learning Research
SFO: A Toolbox for Submodular Function Optimization
The Journal of Machine Learning Research
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HLT '11 Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies: short papers - Volume 2
Parameterized complexity results for exact bayesian network structure learning
Journal of Artificial Intelligence Research
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We show that the class of strongly connected graphical models with tree-width at most k can be properly efficiently PAC-learnt with respect to the Kullback-Leibler Divergence. Previous approaches to this problem, such as those of Chow ([1]), and Hoffgen ([7]) have shown that this class is PAC-learnable by reducing it to a combinatorial optimization problem. However, for k 1, this problem is NP-complete ([15]), and so unless P=NP, these approaches will take exponential amounts of time. Our approach differs significantly from these, in that it first attempts to find approximate conditional independencies by solving (polynomially many) submodular optimization problems, and then using a dynamic programming formulation to combine the approximate conditional independence information to derive a graphical model with underlying graph of the tree-width specified. This gives us an efficient (polynomial time in the number of random variables) PAC-learning algorithm which requires only polynomial number of samples of the true distribution, and only polynomial running time.