Inner Product Spaces Induced By Bayesian Networks And Kernel Functions

The Bayesian networks are graphical representation of multivariate joint probability distribution, which exploit the dependency structure of distributions to describe them in a compact and natural manner. They are always used to find the potential relationship on some data sets. There has been a great deal of research focused on the Bayesian Networks in machine leaning, artificial intelligence, and probability inference due to their uncertain knowledge expression, the ability to match the probability and property for learning based on prior knowledge. It is an important method in the data classification to map the data in the nonlinear separator space to the high-dimension feature space that is separable by Kernel function. In this paper, we try to combine the key advantage of probabilistic models with kernel-based learning systems to present some results for data classification and concept learning.Firstly, we discuss the possibility and several constraint conditions on mapping data to low-dimension feature space as that preserve approximate linear separability and a certain margin value with allowing a certain amount of error. This result provides a theory supported for the upper dimension of inner product spaces induced by Bayesian networks.Secondly, we focus on two-label classification tasks by combining the key advantage of Bayesian networks and kernel methods. We discuss inner product spaces induced by some special unconstraint Bayesian networks over Boolean domain, and give the Euclidean dimension and VC dimension of the inner product spaces of the concept classes induced by them. Based on equivalence property of probability distribution, we mainly study inner product spaces induced by some special unconstraint Bayesian networks with four-valued variables. Further we obtain that the dimension of the inner product spaces of the concept classes induced by them.