VC dimension and inner product space induced by Bayesian networks |
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Authors: | Youlong Yang Yan Wu |
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Affiliation: | aSchool of Science, Xidian University, Xi’an 710071, PR China |
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Abstract: | Bayesian networks are graphical tools used to represent a high-dimensional probability distribution. They are used frequently in machine learning and many applications such as medical science. This paper studies whether the concept classes induced by a Bayesian network can be embedded into a low-dimensional inner product space. We focus on two-label classification tasks over the Boolean domain. For full Bayesian networks and almost full Bayesian networks with n variables, we show that VC dimension and the minimum dimension of the inner product space induced by them are 2n-1. Also, for each Bayesian network we show that if the network constructed from by removing Xn satisfies either (i) is a full Bayesian network with n-1 variables, i is the number of parents of Xn, and i<n-1 or (ii) is an almost full Bayesian network, the set of all parents of Xn PAn={X1,X2,Xn3,…,Xni} and 2i<n-1. Our results in the paper are useful in evaluating the VC dimension and the minimum dimension of the inner product space of concept classes induced by other Bayesian networks. |
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Keywords: | Bayesian networks VC dimension Inner product space Concept classes |
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