Inner Product Space and Concept Classes Induced by Bayesian Networks

Abstract

Bayesian networks have become one of the major models used for statistical inference. In this paper we discuss the properties of the inner product spaces and concept class induced by some special Bayesian networks and the problem whether there exists a Bayesian network such that lower bound on dimensional inner product space just is some positive integer. We focus on two-label classification tasks over the Boolean domain. As main results we show that lower bound on the dimension of the inner product space induced by a class of Bayesian networks without v-structures is $\sum^{n}_{i=1}2^{m_{i}}+1$ where m i denotes the number of parents for ith variable. As the variable�s number of Bayesian network is n?5, we also show that for each integer m?[n+1,2 n -1] there is a Bayesian network $\mathcal{N}$ such that VC dimension of concept class and lower bound on dimensional inner product space induced by $\mathcal{N}$ all are m.