VC dimension and inner product space induced by Bayesian networks

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 X_n satisfies either (i) is a full Bayesian network with n-1 variables, i is the number of parents of X_n, and i<n-1 or (ii) is an almost full Bayesian network, the set of all parents of X_n PA_n={X₁,X₂,X_n3,…,X_ni} 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.