## 1 Introduction  
As one of the top 10 algorithms in the fields of data mining and machine learning [1], the naive Bayesian classifier (NBC) has been used in numerous domains. The main advantage of the NBC is its simple model structure that makes it easy to implement and its good theoretical interpretability. In recent years, the NBC also received much attention from the industry and academia since it can be easily deployed in distributed environments to process big data. Despite possessing several desirable properties, the NBC is built upon a strong assumption, called the attribute independence assumption, which states that condition attributes must be mutually independent with respect to the decision attribute. This assumption simplifies the calculation of posterior probabilities (the probabilities that attribute values of a sample are all observed for a given class). Rather than computing a posterior probability as a joint probability of condition attributes, the NBC calculates it as the product of multiple marginal probabilities. This makes the computation of NBC very efficient and allows the estimation of probabilities even with small data sets. However, the attribute independence assumption does not hold in many real-life data sets, which substantially limits the prediction performance of the NBC. Recent studies on the NBC mainly focused on finding ways to relax the independence assumption so as to further improve its generalization performance. There are two main approaches for improving the testing ability of the NBC [2]: (1) improving the structure of models and (2) performing data transformations. The former approach introduces complex Bayesian network structures to model attribute dependencies, while the latter applies feature selection or feature extraction methods to select independent attributes. Some representative studies of these two approaches are described next. Model structure-oriented improvement methods use different estimation strategies (e.g., density estimation, Bayesian network, and attribute weighting) to estimate the conditional probability that a sample's attribute values are observed given its class. The flexible naive Bayes (FNB) [3] relies on kernel density estimations to determine a class's conditional probabilities. However, FNB can only handle continuous attributes and does not offer a solution to relax the attribute independence assumption. The tree-augmented naive (TAN) Bayes [4] algorithm is a semi-naive Bayesian learning method that relaxes the attribute independence assumption by employing a tree structure, where each condition attribute only depends on the decision attribute and at most one condition attribute. The averaged one-dependent estimator (AODE) [5] is an ensemble classifier that applies an attribute selection algorithm to construct a series of one-dependent classifiers. Each classifier is a simple Bayesian network that is obtained by averaging all one-dependence classifiers. The hidden naive Bayes (HNB) classifier [6] uses the weighted sum of two-attribute dependencies to represent multiple-attribute dependence, where the weights are determined based on the mutual information between two condition attributes. The deep feature weighting for NBC (DFW-NBC) [7] technique and correlation-based feature weighting filter for NBC (CFW-NBC) [8] are two attribute weighting-based NBCs. DFW-NBC estimates each conditional probability of the NBC by deeply computing attribute weighted frequencies from training data while CFW-NBC weights the condition attributes by considering both the mutual relevance and the average mutual redundancy. Data transformation-oriented improvement methods focus on the deep exploration of the training data to avoid complex and time-consuming structure learning. The NBCs obtained by this approach still rely on the independence assumption; thus, the performance may be degraded if the training data do not satisfy that assumption.