Lazy Learning of Bayesian Rules

The naive Bayesian classifier provides a simple and effective approach to classifier learning, but its attribute independence assumption is often violated in the real world. A number of approaches have sought to alleviate this problem. A Bayesian tree learning algorithm builds a decision tree, and generates a local naive Bayesian classifier at each leaf. The tests leading to a leaf can alleviate attribute inter-dependencies for the local naive Bayesian classifier. However, Bayesian tree learning still suffers from the small disjunct problem of tree learning. While inferred Bayesian trees demonstrate low average prediction error rates, there is reason to believe that error rates will be higher for those leaves with few training examples. This paper proposes the application of lazy learning techniques to Bayesian tree induction and presents the resulting lazy Bayesian rule learning algorithm, called LBR. This algorithm can be justified by a variant of Bayes theorem which supports a weaker conditional attribute independence assumption than is required by naive Bayes. For each test example, it builds a most appropriate rule with a local naive Bayesian classifier as its consequent. It is demonstrated that the computational requirements of LBR are reasonable in a wide cross-section of natural domains. Experiments with these domains show that, on average, this new algorithm obtains lower error rates significantly more often than the reverse in comparison to a naive Bayesian classifier, C4.5, a Bayesian tree learning algorithm, a constructive Bayesian classifier that eliminates attributes and constructs new attributes using Cartesian products of existing nominal attributes, and a lazy decision tree learning algorithm. It also outperforms, although the result is not statistically significant, a selective naive Bayesian classifier.     

Naive Bayes text classifiers: a locally weighted learning approach

Due to being fast, easy to implement and relatively effective, some state-of-the-art naive Bayes text classifiers with the strong assumption of conditional independence among attributes, such as multinomial naive Bayes, complement naive Bayes and the one-versus-all-but-one model, have received a great deal of attention from researchers in the domain of text classification. In this article, we revisit these naive Bayes text classifiers and empirically compare their classification performance on a large number of widely used text classification benchmark datasets. Then, we propose a locally weighted learning approach to these naive Bayes text classifiers. We call our new approach locally weighted naive Bayes text classifiers (LWNBTC). LWNBTC weakens the attribute conditional independence assumption made by these naive Bayes text classifiers by applying the locally weighted learning approach. The experimental results show that our locally weighted versions significantly outperform these state-of-the-art naive Bayes text classifiers in terms of classification accuracy.     

Maximum margin Bayesian network classifiers

We present a maximum margin parameter learning algorithm for Bayesian network classifiers using a conjugate gradient (CG) method for optimization. In contrast to previous approaches, we maintain the normalization constraints on the parameters of the Bayesian network during optimization, i.e., the probabilistic interpretation of the model is not lost. This enables us to handle missing features in discriminatively optimized Bayesian networks. In experiments, we compare the classification performance of maximum margin parameter learning to conditional likelihood and maximum likelihood learning approaches. Discriminative parameter learning significantly outperforms generative maximum likelihood estimation for naive Bayes and tree augmented naive Bayes structures on all considered data sets. Furthermore, maximizing the margin dominates the conditional likelihood approach in terms of classification performance in most cases. We provide results for a recently proposed maximum margin optimization approach based on convex relaxation. While the classification results are highly similar, our CG-based optimization is computationally up to orders of magnitude faster. Margin-optimized Bayesian network classifiers achieve classification performance comparable to support vector machines (SVMs) using fewer parameters. Moreover, we show that unanticipated missing feature values during classification can be easily processed by discriminatively optimized Bayesian network classifiers, a case where discriminative classifiers usually require mechanisms to complete unknown feature values in the data first.     

基于特征加权的朴素贝叶斯分类器

朴素贝叶斯分类器是一种广泛使用的分类算法,其计算效率和分类效果均十分理想。但是,由于其基础假设“朴素贝叶斯假设”与现实存在一定的差异,因此在某些数据上可能导致较差的分类结果。现在存在多种方法试图通过放松朴素贝叶斯假设来增强贝叶斯分类器的分类效果,但是通常会导致计算代价大幅提高。该文利用特征加权技术来增强朴素贝叶斯分类器。特征加权参数直接从数据导出,可以看作是计算某个类别的后验概率时,某个属性对于该计算的影响程度。数值实验表明,特征加权朴素贝叶斯分类器（FWNB）的效果与其他的一些常用分类算法,例如树扩展朴素贝叶斯（TAN）和朴素贝叶斯树（NBTree）等的分类效果相当,其平均错误率都在17％左右;在计算速度上,FWNB接近于NB,比TAN和NBTree快至少一个数量级。     

An intrusion detection and alert correlation approach based on revising probabilistic classifiers using expert knowledge

Bayesian networks are important knowledge representation tools for handling uncertain pieces of information. The success of these models is strongly related to their capacity to represent and handle dependence relations. Some forms of Bayesian networks have been successfully applied in many classification tasks. In particular, naive Bayes classifiers have been used for intrusion detection and alerts correlation. This paper analyses the advantage of adding expert knowledge to probabilistic classifiers in the context of intrusion detection and alerts correlation. As examples of probabilistic classifiers, we will consider the well-known Naive Bayes, Tree Augmented Naïve Bayes (TAN), Hidden Naive Bayes (HNB) and decision tree classifiers. Our approach can be applied for any classifier where the outcome is a probability distribution over a set of classes (or decisions). In particular, we study how additional expert knowledge such as “it is expected that 80 % of traffic will be normal” can be integrated in classification tasks. Our aim is to revise probabilistic classifiers’ outputs in order to fit expert knowledge. Experimental results show that our approach improves existing results on different benchmarks from intrusion detection and alert correlation areas.     

广义朴素贝叶斯分类器

朴素贝叶斯分类器具有很高的学习和分类效率,但不能充分利用属性变量之间的依赖信息.贝叶斯网络分类器具有很强的分类能力,但分类器学习比较复杂.本文建立广义朴素贝叶斯分类器,它具有灵活的分类能力选择方式、效率选择方式及学习方式,能够弥补朴素贝叶斯分类器和贝叶斯网络分类器的不足,并继承它们的优点.     

Multiple Bayesian discriminant functions for high-dimensional massive data classification

The presence of complex distributions of samples concealed in high-dimensional, massive sample-size data challenges all of the current classification methods for data mining. Samples within a class usually do not uniformly fill a certain (sub)space but are individually concentrated in certain regions of diverse feature subspaces, revealing the class dispersion. Current classifiers applied to such complex data inherently suffer from either high complexity or weak classification ability, due to the imbalance between flexibility and generalization ability of the discriminant functions used by these classifiers. To address this concern, we propose a novel representation of discriminant functions in Bayesian inference, which allows multiple Bayesian decision boundaries per class, each in its individual subspace. For this purpose, we design a learning algorithm that incorporates the naive Bayes and feature weighting approaches into structural risk minimization to learn multiple Bayesian discriminant functions for each class, thus combining the simplicity and effectiveness of naive Bayes and the benefits of feature weighting in handling high-dimensional data. The proposed learning scheme affords a recursive algorithm for exploring class density distribution for Bayesian estimation, and an automated approach for selecting powerful discriminant functions while keeping the complexity of the classifier low. Experimental results on real-world data characterized by millions of samples and features demonstrate the promising performance of our approach.     

Boosted Bayesian network classifiers

The use of Bayesian networks for classification problems has received a significant amount of recent attention. Although computationally efficient, the standard maximum likelihood learning method tends to be suboptimal due to the mismatch between its optimization criteria (data likelihood) and the actual goal of classification (label prediction accuracy). Recent approaches to optimizing classification performance during parameter or structure learning show promise, but lack the favorable computational properties of maximum likelihood learning. In this paper we present boosted Bayesian network classifiers, a framework to combine discriminative data-weighting with generative training of intermediate models. We show that boosted Bayesian network classifiers encompass the basic generative models in isolation, but improve their classification performance when the model structure is suboptimal. We also demonstrate that structure learning is beneficial in the construction of boosted Bayesian network classifiers. On a large suite of benchmark data-sets, this approach outperforms generative graphical models such as naive Bayes and TAN in classification accuracy. Boosted Bayesian network classifiers have comparable or better performance in comparison to other discriminatively trained graphical models including ELR and BNC. Furthermore, boosted Bayesian networks require significantly less training time than the ELR and BNC algorithms.     

Combining Classifiers with Meta Decision Trees

The paper introduces meta decision trees (MDTs), a novel method for combining multiple classifiers. Instead of giving a prediction, MDT leaves specify which classifier should be used to obtain a prediction. We present an algorithm for learning MDTs based on the C4.5 algorithm for learning ordinary decision trees (ODTs). An extensive experimental evaluation of the new algorithm is performed on twenty-one data sets, combining classifiers generated by five learning algorithms: two algorithms for learning decision trees, a rule learning algorithm, a nearest neighbor algorithm and a naive Bayes algorithm. In terms of performance, stacking with MDTs combines classifiers better than voting and stacking with ODTs. In addition, the MDTs are much more concise than the ODTs and are thus a step towards comprehensible combination of multiple classifiers. MDTs also perform better than several other approaches to stacking.     

Bayesian network classifiers for mineral potential mapping

In this paper, we describe three Bayesian classifiers for mineral potential mapping: (a) a naive Bayesian classifier that assumes complete conditional independence of input predictor patterns, (b) an augmented naive Bayesian classifier that recognizes and accounts for conditional dependencies amongst input predictor patterns and (c) a selective naive classifier that uses only conditionally independent predictor patterns. We also describe methods for training the classifiers, which involves determining dependencies amongst predictor patterns and estimating conditional probability of each predictor pattern given the target deposit-type. The output of a trained classifier determines the extent to which an input feature vector belongs to either the mineralized class or the barren class and can be mapped to generate a favorability map. The procedures are demonstrated by an application to base metal potential mapping in the proterozoic Aravalli Province (western India). The results indicate that although the naive Bayesian classifier performs well and shows significant tolerance for the violation of the conditional independence assumption, the augmented naive Bayesian classifier performs better and exhibits finer generalization capability. The results also indicate that the rejection of conditionally dependent predictor patterns degrades the performance of a naive classifier.     

Bayesian network classifiers based on Gaussian kernel density

For learning a Bayesian network classifier, continuous attributes usually need to be discretized. But the discretization of continuous attributes may bring information missing, noise and less sensitivity to the changing of the attributes towards class variables. In this paper, we use the Gaussian kernel function with smoothing parameter to estimate the density of attributes. Bayesian network classifier with continuous attributes is established by the dependency extension of Naive Bayes classifiers. We also analyze the information provided to a class for each attributes as a basis for the dependency extension of Naive Bayes classifiers. Experimental studies on UCI data sets show that Bayesian network classifiers using Gaussian kernel function provide good classification accuracy comparing to other approaches when dealing with continuous attributes.     

A Novel Bayes Model: Hidden Naive Bayes

Because learning an optimal Bayesian network classifier is an NP-hard problem, learning-improved naive Bayes has attracted much attention from researchers. In this paper, we summarize the existing improved algorithms and propose a novel Bayes model: hidden naive Bayes (HNB). In HNB, a hidden parent is created for each attribute which combines the influences from all other attributes. We experimentally test HNB in terms of classification accuracy, using the 36 UCI data sets selected by Weka, and compare it to naive Bayes (NB), selective Bayesian classifiers (SBC), naive Bayes tree (NBTree), tree-augmented naive Bayes (TAN), and averaged one-dependence estimators (AODE). The experimental results show that HNB significantly outperforms NB, SBC, NBTree, TAN, and AODE. In many data mining applications, an accurate class probability estimation and ranking are also desirable. We study the class probability estimation and ranking performance, measured by conditional log likelihood (CLL) and the area under the ROC curve (AUC), respectively, of naive Bayes and its improved models, such as SBC, NBTree, TAN, and AODE, and then compare HNB to them in terms of CLL and AUC. Our experiments show that HNB also significantly outperforms all of them.     

Learning Bayesian classifiers from positive and unlabeled examples

The positive unlabeled learning term refers to the binary classification problem in the absence of negative examples. When only positive and unlabeled instances are available, semi-supervised classification algorithms cannot be directly applied, and thus new algorithms are required. One of these positive unlabeled learning algorithms is the positive naive Bayes (PNB), which is an adaptation of the naive Bayes induction algorithm that does not require negative instances. In this work we propose two ways of enhancing this algorithm. On one hand, we have taken the concept behind PNB one step further, proposing a procedure to build more complex Bayesian classifiers in the absence of negative instances. We present a new algorithm (named positive tree augmented naive Bayes, PTAN) to obtain tree augmented naive Bayes models in the positive unlabeled domain. On the other hand, we propose a new Bayesian approach to deal with the a priori probability of the positive class that models the uncertainty over this parameter by means of a Beta distribution. This approach is applied to both PNB and PTAN, resulting in two new algorithms. The four algorithms are empirically compared in positive unlabeled learning problems based on real and synthetic databases. The results obtained in these comparisons suggest that, when the predicting variables are not conditionally independent given the class, the extension of PNB to more complex networks increases the classification performance. They also show that our Bayesian approach to the a priori probability of the positive class can improve the results obtained by PNB and PTAN.     

基于朴素Bayes组合的简易集成分类器

朴素Bayes分类器是一种简单有效的机器学习工具.本文用朴素Bayes分类器的原理推导出"朴素Bayes组合"公式,并构造相应的分类器.经过测试,该分类器有较好的分类性能和实用性,克服了朴素Bayes分类器精确度差的缺点,并且比其他分类器更加快速而不会显著丧失精确度.     

一种限定性的双层贝叶斯分类模型

朴素贝叶斯分类模型是一种简单而有效的分类方法,但它的属性独立性假设使其无法表达属性变量间存在的依赖关系,影响了它的分类性能.通过分析贝叶斯分类模型的分类原则以及贝叶斯定理的变异形式,提出了一种基于贝叶斯定理的新的分类模型DLBAN(double-level Bayesian network augmented naive Bayes).该模型通过选择关键属性建立属性之间的依赖关系.将该分类方法与朴素贝叶斯分类器和TAN(tree augmented naive Bayes)分类器进行实验比较.实验结果表明,在大多数数据集上,DLBAN分类方法具有较高的分类正确率.     

A ‘non-parametric’ version of the naive Bayes classifier

Many algorithms have been proposed for the machine learning task of classification. One of the simplest methods, the naive Bayes classifier, has often been found to give good performance despite the fact that its underlying assumptions (of independence and a normal distribution of the variables) are perhaps violated. In previous work, we applied naive Bayes and other standard algorithms to a breast cancer database from Nottingham City Hospital in which the variables are highly non-normal and found that the algorithm performed well when predicting a class that had been derived from the same data. However, when we then applied naive Bayes to predict an alternative clinical variable, it performed much worse than other techniques. This motivated us to propose an alternative method, based on naive Bayes, which removes the requirement for the variables to be normally distributed, but retains the essential structure and other underlying assumptions of the method. We tested our novel algorithm on our breast cancer data and on three UCI datasets which also exhibited strong violations of normality. We found our algorithm outperformed naive Bayes in all four cases and outperformed multinomial logistic regression (MLR) in two cases. We conclude that our method offers a competitive alternative to MLR and naive Bayes when dealing with data sets in which non-normal distributions are observed.     

基于贝叶斯学习的集成流量分类方法

NB方法条件独立性假设和BAN方法小训练集难以建模。为此,提出一种基于贝叶斯学习的集成流量分类方法。构造单独的NB和BAN分类器,在此基础上利用验证集得到各分类器的权重,通过加权平均组合各分类器的输出,实现网络流量分类。以Moore数据集为实验数据,并与NB方法和BAN方法相比较,结果表明,该方法具有更高的分类准确率和稳定性。     

一种属性相关性的加权贝叶斯分类算法研究

根据RoughSet属性重要度理论,构建了基于互信息的属性子集重要度,提出属性相关性的加权朴素贝叶斯分类算法,该算法同时放宽了朴素贝叶斯算法属性独立性、属性重要性相同的假设。通过在UCI部分数据集上进行仿真实验,与基于属性相关性分析的贝叶斯(CB)和加权朴素贝叶斯(WNB)两种算法做比较,证明了该算法的有效性。     

Double-layer bayesian classifier ensembles based on frequent itemsets

Numerous models have been proposed to reduce the classification error of Na¨ ve Bayes by weakening its attribute independence assumption and some have demonstrated remarkable error performance. Considering that ensemble learning is an effective method of reducing the classification error of the classifier, this paper proposes a double-layer Bayesian classifier ensembles (DLBCE) algorithm based on frequent itemsets. DLBCE constructs a double-layer Bayesian classifier (DLBC) for each frequent itemset the new instance contained and finally ensembles all the classifiers by assigning different weight to different classifier according to the conditional mutual information. The experimental results show that the proposed algorithm outperforms other outstanding algorithms.