Bayesian Network Classifiers in a High Dimensional Framework

We present a growing dimension asymptotic formalism. The perspective in this paper is classification theory and we show that it can accommodate probabilistic networks classifiers, including naive Bayes model and its augmented version. When represented as a Bayesian network these classifiers have an important advantage: The corresponding discriminant function turns out to be a specialized case of a generalized additive model, which makes it possible to get closed form expressions for the asymptotic misclassification probabilities used here as a measure of classification accuracy. Moreover, in this paper we propose a new quantity for assessing the discriminative power of a set of features which is then used to elaborate the augmented naive Bayes classifier. The result is a weighted form of the augmented naive Bayes that distributes weights among the sets of features according to their discriminative power. We derive the asymptotic distribution of the sample based discriminative power and show that it is seriously overestimated in a high dimensional case. We then apply this result to find the optimal, in a sense of minimum misclassification probability, type of weighting.