VC dimension and inner product space induced by Bayesian networks

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 N we show that VCdim(N)=Edim(N)=2n-1+2i if the network N′ constructed from N by removing Xn satisfies either (i) N′ is a full Bayesian network with n-1 variables, i is the number of parents of Xn, and i<n-1 or (ii) N′ is an almost full Bayesian network, the set of all parents of XnPAn={X1,X2,Xn3,…,Xni} and 2⩽i<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.