Bayesian Networks : A State-Of-The-Art Survey Nelda Kote Marenglen Biba Elena Canaj Department of Fundamentals Department of Computer Department of Computer of Computer Science, Faculty Science, Faculty of Engineering, Faculty of of Information Technology, Information Technology, Information Technology, Polytechnic University of New York University of Polytechnic University of Tirana, Albania Tirana, Albania Tirana, Albania nkote@fti.edu.al marenglenbiba@unyt.edu.al elenacanaj@gmail.com in terms of their model source. Also, due to their Abstract graphical structure, machine-learned Bayesian Networks are intuitively interpretable, thus facilitating Over the last decade, Bayesian Networks human learning and theory building. Bayesian (BNs) have become an increasingly popular Networks allow human learning and machine learning Artificial Intelligence approach. BNs are a to interact efficiently. This way, Bayesian Networks widely used method in the modelling of can be developed from a combination of human and uncertain knowledge. There have been many artificial intelligence. important new developments in this field. This paper presents a review and classification scheme for recent researches on Bayesian Networks. This is achieved by reviewing relevant articles published in the recent years. The articles are classified based on a scheme that consists of three main Bayesian Networks topics: Bayesian Networks Structure Learning, Advanced Application of Bayesian Networks and Bayesian Network Classifiers. This review provides a reference source and classification scheme for researchers interested in BNs, and indicates under-researched areas as well as future directions. 1 Introduction This paper presents a review of recent researches in the area of Bayesian Networks. BNs are a popular class of probabilistic graphical models for researches and applications in the field of Artificial Intelligence. BNs are built on Bayes’ theorem and allow to represent a joint probability distribution over a set of variables in Figure 1: Bayesian Networks spanning theory and the network. In Bayesian probabilistic inference, the data [Con13] joint distribution over the set of variables in a Bayesian Network can be used to calculate the probabilities of Figure 1 illustrates the role and position of any configuration of these variables given fixed values Bayesian Networks between theory and data in of another set of variables, called observations or Artificial Intelligence. This paper addresses most of the evidence [Rus09]. recent research works of three main Bayesian Bayesian Networks can be built from human Networks fields: Bayesian Networks Structure knowledge, i.e. from theory, or, they can be machine Learning, Advanced Application of Bayesian learned from data. Thus, they cover the entire spectrum Networks and Bayesian Networks Classifiers. The structure of the paper is as follows: Section 2 Based on the classification scheme, we give the presents the research methodology; Section 3 presents results on the total number of publications per domain results and analysis of the searches in a quantitative and their percentage on the total numbers of reviewed perspective; Section 4 gives the detailed description papers, shown in table 1. The results that we found is and evaluation of the reviewed papers and finally we that researches are equally focused on these three main conclude our work in Section 5. BNs fields. 2 Methodology Table 1: Distribution of covered network aspects The scope of this review is to identify and evaluate the Total no. of Percentage Papers / recent research fields on Bayesian Networks. Over fifty Domain publications Domain (%) papers were first extracted from searches made on three BNs Structure major research databases for computer science: IEEE 9 30 Learning Xplore, CiteSeerX and Google Scholar, for the following keywords: Bayesian Networks, data Application of 10 33 classification, learning structure, data mining, Bayes BNs Theorem. The date range for this search was limited BNs Classifiers 11 37 from 2011 until 2018. We kept our scope wider to consider all topics of Bayesian Networks. The Total 30 100 challenges related to the structure learning methods and algorithms, implementation of different applications and classification methods and algorithms were all Table 2 shows the total number of publications per within the scope of this review paper. The citation- year and their percentage on the total numbers of references of the selected papers were checked, and reviewed papers. additional papers were found to be necessary to add to this review based on the criteria mentioned above. From the numerous research publications, around thirty Table 2: Representation of the total number of papers were selected for this review. publications per year The papers are categorized based on their main focus in three groups: Bayesian Networks Structure Learning, Advanced Application of Bayesian Networks Total nr. of Percentage Papers Year and Bayesian Networks Classifiers. publications /Year (%) 2011 3 10 3 Literature Review: Quantitative Results 2012 2 7 and Analysis 2013 4 13 In this section we present the results of our study, based 2014 4 13 on the methodology explained in section 2. All thirty 2015 6 20 selected publications are analyzed and evaluated based 2016 3 10 on their research contributions. The articles are noted 2017 5 17 by their type as Review, Survey, Improvements in 2018 3 10 existing Technology, New Proposal and special attention is given to real experiments, Total 30 100 simulation/emulation and system implementation made by authors. Table 3 shows all the selected papers for this review. Table 3: List of articles Article BNs Application of BNs Article Type Measures / Learning BNs Classifiers Experiments Structure [Kos12] Main Topic X Review [Dal11] Main Topic X Review/Improvements [Mal15] Main Topic X Review/Improvements X [Zha14] Main Topic X Improvements/New Design X [Mil15] Main Topic X New Design X [Tsc15] Main Topic X X New Design X [Li17] Main Topic X New Design X [Kar16] Main Topic X New Design X [Zha18] Main Topic X New Design X [Per14] Main Topic Improvements/New Design X [Yua11] X Main Topic Review/Improvements X [Vle15] X Main Topic New Design X [Oku12] Main Topic Review/Improvements [Kle15] X Main Topic Review/Improvements X [Cay11] Main Topic New Design X [Lan13] Main Topic Review/Improvements X [Ren13] Main Topic New Design X [Urs17] X Main Topic New Design X [Wee18] Main Topic New Design X [Bie14] X Main Topic Survey [Ang16] X X Main Topic New Design X [Vij13] Main Topic Comparative Analysis X [Suc14] X Main Topic New Design X [Cho16] X Main Topic Review/Improvements X [Liu13] X Main Topic New Design X [Tsc15] X Main Topic New Design X [Xu17] X Main Topic New Design X [Ans17] X Main Topic Improvements X [Kan17] X Main Topic Improvements X [Wu18] X Main Topic Improvements/Comparative X Analysis 4.2 Bayesian Networks Structure Learning 4 Literature Review: Topics-Related In this section we review the recent research works in Analysis Bayesian Networks structure learning and analyze their characteristics. We have reviewed nine papers in This section is an overview of each of the domains. terms of Bayesians Network Structure Learning. The 30 publications are mapped based on main topic Bayesian Networks Structure Learning problem and their contributions on Bayesian Networks, as well takes the data as input and produces a directed acyclic as references and possible analysis. First, we give a graph as the output. There are roughly three main general description of Bayesian Networks. approaches to the learning problem: score-based learning, constraint-based learning, and hybrid 4.1 The Bayesian Network methods. These approaches are reviewed in detail in A Bayesian Network is a form of probabilistic three papers [Kos12], [Dal11], [Mal15]. Score-based graphical model. Structurally, a Bayesian Network is learning methods evaluate the quality of Bayesian a directed acyclic graph where nodes represent Network structures using a scoring function and select variables and arcs represent dependency relations the one that has the best score. These methods between the variables (nodes). An arc from node A to basically formulate the learning problem as a another node B is called: A is a parent of B. A node combinatorial optimization problem. They work well can represent any kind of random variable. for datasets with not too many variables but may fail A Bayesian network with parameters is a graphical to find optimal solutions for large datasets. representation of the joint distribution over all the Constraint-based learning methods typically use variables represented by nodes in the graph. If the statistical tests to identify conditional independence variables are X1,..., Xn we let “parents(A)” be the relations from the data and build a Bayesian Network parents of the node A. Then the joint distribution for structure that best fits those independence relations. X1 through Xn is represented as the product of the Constraint-based methods mostly rely on results of probability distributions: local statistical tests, so they can often scale to large datasets. However, they are sensitive to the accuracy P(X1, ... , Xn ) = P(Xi parents (Xi)) for i = 1 to n. of the statistical tests and may not work well when there are insufficient or noisy data. In comparison, To fully specify the Bayesian Network and to carry score-based methods work well even for datasets with out numerical calculations, it is necessary to further relatively few data points. Hybrid methods aim to specify for each node X the probability distribution integrate the advantages of the previous two for X conditional on its parents. In this way a approaches and use combinations of constraint-based Bayesian Network could be used to perform any and/or score-based methods for solving the learning probabilistic inference over the domain variables problem. One popular strategy is to use constraint- [Rus09]. based learning to create a skeleton graph and then use Important usage of Bayesian Networks is made in score-based learning to find a high-scoring network modeling, where the structure of the Bayesian structure that is a subgraph of the skeleton. network is generated by software. Learning the Authors in [Kos12] and [Dal11] take a broad look structure of a Bayesian Network is a very important at the literature on learning Bayesian Networks in task in machine learning. To find the structure of the particular their structure from data. network, a scoring function should be maximized Authors in [Mal15] present results from an through a search algorithm. We review this topic in empirical evaluation of the impact of Bayesian section 4.2. Network structure learning strategies on the learned Bayesian Networks are used for modeling structures. They investigate how learning algorithms knowledge in many domains with uncertain with different optimality guarantees compare in terms knowledge, like medicine, engineering, text analysis, of structural aspects and generalizability of the image processing, data fusion, decision support produced network structures. systems, and data classification. The recent researches Articles [Zha14], [Mil15], [Tsc15], [Li17], on these topics are reviewed in sections 4.3 and 4.4. [Kar16], [Zha18] give further details on learning structures and evaluate algorithms used for data among which DE operators are adopted to lead the learning. evolutionary process. Experimental results show that Authors in [Zha14] aim to provide a timely review this algorithm outperforms the basic ACO in learning on this area with emphasis on state-of-the-art multi- BN structure in terms of convergence and accuracy. label learning algorithms. Firstly, fundamentals on At the end we observed that score-based exact multi-label learning including formal definition and structure learning has become an active research topic evaluation metrics are given. Secondly and primarily, in recent years. In this context, a scoring function is eight representative multi-label learning algorithms used to measure the goodness of the data fitting a are scrutinized under common notations with relevant structure. The goal is to find the structure which analyses and discussions. Thirdly, several related optimizes the scoring function, and it has been shown learning settings are briefly summarized. a NP-hard problem. In the work presented in [Mil15] a set of experiments are performed to compare the 4.3 Application of Bayesian Networks performance of two Bayesian Student Models, whose parameters have been specified by experts and learnt Bayesian Networks are used for modeling knowledge from data respectively. Results show that both models in many domains with uncertain knowledge, like are able to provide reasonable estimations for medicine, engineering, text analysis, image knowledge variables in the student model, in spite of processing, data fusion, decision support systems, and the small size of the dataset available for learning the data classification. Ten papers that address different parameters. application of BN are reviewed in this section. Article [Tsc15] presents generative and The first article of this domain [Per14], presents an discriminative learning algorithms for Bayesian approach to directly infer individual differences network classifiers relying only on reduced-precision related to subjective mental representations within the arithmetic. For several standard benchmark datasets, framework of Bayesian models of cognition. In this these algorithms achieve classification-rate approach, Bayesian data analysis methods are used to performance close to that of Bayesian Network estimate cognitive parameters and motivate the classifiers with parameters learned by conventional inference process within a Bayesian cognitive model. algorithms using double precision floating-point Authors illustrate this integrative Bayesian approach arithmetic. on a model of memory. They apply the model to Authors in [Li17] by combining the advantages of behavioral data from a memory experiment involving constraint-based and score-based algorithms, the recall of heights of people. A cross-validation proposed a hybrid distributed Bayesian Network analysis shows that the Bayesian memory model with structure learning algorithm from large-scale dataset inferred subjective priors predicts withheld data better using MapReduce. The algorithm reuses the statistical than a Bayesian model where the priors are based on results of MapReduce that makes it possible for environmental statistics. In addition, the model with learning structures accurately. The experimental inferred priors at the individual subject level led to the results show that the proposed solution has good best overall generalization performance, suggesting results in both efficiency and accuracy. that individual differences are important to consider In [Kar16], the authors proposed a new approach in Bayesian models of cognition. to accelerate the exact structure learning of Bayesian Authors in [Yua11] introduce a method called Networks. This approach leverages relationship Most Relevant Explanation (MRE) which finds a between a partial network structure and the remaining partial instantiation of the target variables that variables to constrain the number of ways in which maximizes the generalized Bayes factor (GBF) as the the partial network can be optimally extended. best explanation for the given evidence. This study Experimental results show that the proposed method shows that GBF has several theoretical properties that performs extremely well in practice, even though it enable MRE to automatically identify the most does not improve the worst-case complexity. relevant target variables in forming its explanation. In Authors in [Zha18] present a new algorithm for particular, conditional Bayes factor (CBF), defined as learning BNs based on the hybrid ACO and the GBF of a new explanation conditioned on an differential evolution (DE). In this hybrid algorithm, existing explanation, provides a soft measure on the the entire ant colony is divided into different groups, degree of relevance of the variables in the new explanation in explaining the evidence given the emotions in the document by establishing a existing explanation. As a result, MRE is able to relationship between the topic modeling and automatically prune less relevant variables from its analyzing the emotions. The experimental results explanation. Authors show that CBF is able to capture show that the proposed method has good performance well the explaining-away phenomenon that is often and can be used in complex domains. represented in Bayesian networks. Moreover, they Authors in [Urs17] proposed to use a Bayesian define two dominance relations between the candidate networks mathematical model to evaluate the software solutions and use the relations to generalize MRE to quality, from the reliability point of view. This model find a set of top explanations that is both diverse and evaluates the reliability of a software system for EMS representative. Case studies on several benchmark (Energy Management Systems) and DMS diagnostic Bayesian networks show that MRE is often (Distribution Management System) that are the core able to find explanatory hypotheses that are not only of national energy system as they are used the precise but also concise. National Dispatch Control Center. To evaluate the The article [Vle15] proposes to combine Bayesian performance of the proposed approach the authors Networks with a narrative approach to reasoning with perform a simulation to obtain some practical results legal evidence, the result of which allows a juror to and draw important conclusions if this model can reason with alternative scenarios while also improve the EMS and DMS software systems. incorporating probabilistic information. The proposed In [Wee18] is used a combination of Bayesian method aids both the construction and the Network and fuzzy cognitive maps (FCM) for understanding of Bayesian networks, using scenario modeling and analyzing network intrusions. First, the schemes. BN is learnt from network intrusion data; following Authors in [Oku12] use Bayesian Networks to this, an FCM is generated from the BN, using a determine the probabilistic influential relationships migration method. The proposed method of network among software metrics and defect proneness. intrusion analysis using both BN and FCM consists of In [Kle15] authors have made a systematic review several stages, in order to leverage the capabilities of that investigates the psychometric analysis of each approach in building the causal model and performance data of simulation-based assessment performing causal analysis. (SBA) and game-based assessment (GBA). The application of Bayesian networks for modeling In [Cay11], Bayesian networks are used to extract knowledge domain is most researched on recent years. the effects of data mining algorithm parameters on the 50% of our reviewed papers, implement and propose final model obtained, both in terms of efficiency and new designs for modeling knowledge in many efficacy in a given situation. Based on this domains with uncertain knowledge. knowledge, authors propose to infer future algorithm configurations appropriate for situations. Instantiation 4.4 Bayesian Network Classifiers of the approach for association rules is also shown in the paper and the feasibility of the approach is This section reviews the theory and validated by the experimentation. implementation of Bayesian Networks in the context Authors in [Lan13] review several BN-based of classification. Bayesian networks provide a very ecosystem service (ESS) models developed in the last general and yet effective graphical language for decade. A SWOT analysis highlights the advantages factoring joint probability distributions which in turn and disadvantages of BNs in ESS modelling and make them very popular for classification. pinpoints remaining challenges for future research. Figure 2 depicts the possible structure of a The existing BN models are suited to describe, Bayesian network used for classification. The dotted analyze, predict and value ESS. Nevertheless, some lines denote potential links, and the blue box is used weaknesses must be considered, including poor to indicate that additional nodes and links can be flexibility of frequently applied software packages, added to the model, usually between the input and difficulties in eliciting expert knowledge and the output nodes. In order to perform classification with a inability to model feedback loops. Bayesian Network such as the one depicted in Figure In [Ren13] the authors used a hierarchical 2, first evidence must be set on the input nodes, and Bayesian network to build a model for the analysis of then the output nodes can be queried using standard the human beings’ emotions. It finds complex Bayesian network inference. The result will be a distribution for each output node, so that you can not Bayes. In lazy classifier has three algorithms namely only determine the most probable state for each IBL, IBK and Kstar. The performances of Bayesian output, but also see the probability assigned to each and lazy classifiers are analyzed by applying various output state. [Xu13] performance factors. From the experimental results, it is observed that the lazy classifier is more efficient than Bayesian classifier. In [Suc14] authors introduce a method for chaining Bayesian classifiers that combines the strengths of classifier chains and Bayesian networks for multi- label classification. A Bayesian Network is induced from data to represent the probabilistic dependency relationships between classes, constrain the number of class variables used in the chain classifier by considering conditional independence conditions, and reduce the number of possible chain orders. The effects in the Bayesian chain classifier performance of considering different chain orders, training strategies, number of class variables added in the base classifiers, and different base classifiers, are experimentally assessed. In particular, it is shown that a random chain order considering the constraints imposed by a Bayesian Network with a simple tree- based structure can have very competitive results in terms of predictive performance and time complexity Figure 2: Generic structure of a Bayesian Network against related state-of the art approaches. classifier [Xu13] Authors in [Cho16] propose the structured Naive Bayes (SNB) classifier, which augments the Authors in [Bie14] survey the whole set of discrete ubiquitous Naive Bayes classifier with structured Bayesian Network classifiers devised to date, features. SNB classifiers facilitate the use of complex organized in increasing order of structure complexity: features, such as combinatorial objects (e.g., graphs, Naive Bayes, selective Naive Bayes, Seminaive paths and orders) in a general but systematic way. Bayes, One Dependence Bayesian classifiers, k- Underlying the SNB classifier is the recently dependence Bayesian classifiers, Bayesian network- proposed Probabilistic Sentential Decision Diagram augmented naive Bayes, Markov blanket-based (PSDD), which is a tractable representation of Bayesian classifier, unrestricted Bayesian classifiers, probability distributions over structured spaces. They and Bayesian multinets. Issues of feature subset illustrate the utility and generality of the SNB selection and generative and discriminative structure classifier via case studies. First, they show how to and parameter learning are also covered. distinguish players of simple games in terms of play In [Ang16], the authors show the accuracy of a style and skill level based purely on observing the General Bayesian Network (GBN) used with the Hill- games they play. Second, they show how to detect Climbing learning method, which does not impose anomalous paths taken on graphs based purely on any restrictions on the structure and better represents observing the paths themselves. the dataset. The results show that it gives equivalent In paper [Liu13], the scalability of Naıve Bayes performances or even outperforms Naive Bayes and classifier (NBC) is evaluated in large datasets. Instead Tree Augmented Naive Bayes in most of the data of using a standard library (e.g., Mahout), authors classification. implemented NBC to achieve fine-grain control of the In the research work of [Vij13], authors have analysis procedure. A Big Data analyzing system is analyzed the performance of Bayesian and Lazy also design for this study. The result is encouraging in classifiers for classifying the files which are stored in that the accuracy of NBC is improved and approaches the computer hard disk. There are two algorithms in 82% when the dataset size increases. The authors Bayesian classifier namely BayesNet, and Naïve have demonstrated that NBC is able to scale up to analyze the sentiment of millions movie reviews with algorithm to obtain the optimal BN classifier for point increasing throughput. cloud classification. Experiment results show that the In [Tsc15], authors investigate the effect of BN classifier can effectively distinguish four types of precision reduction of the parameters on the basic ground objects, including ground, vegetation, classification performance of Bayesian Network trees, and buildings, with a high accuracy. Moreover, classifiers (BNCs). The probabilities are either compared with other classifiers, the proposed BN determined generatively or discriminatively. classifier can achieve the highest overall accuracies, Discriminative probabilities are typically more and in particular, the classifier demonstrates its extreme. However, the results indicate that BNCs advantage in the classification of ground and low with discriminatively optimized parameters are almost vegetation points. as robust to precision reduction as BNCs with Authors in [Wu18] to improve the safety of bus generatively optimized parameters. Furthermore, even driving, classify the specific types of latent abnormal large precision reduction does not decrease driving behavior, which include sudden braking, lane classification performance significantly. These results changing casually, quick turn, fast U-turn and allow the implementation of BNCs with less longtime parking, and propose a method to identify computational complexity. This supports application the abnormal driving behavior of the bus. After in embedded systems using floating-point numbers collecting the data, they extract features in thirteen with small bit-width. Reduced bit-widths further dimensions and then train the Naive Bayesian enable to represent BNCs in the integer domain while classifier, which is employed to detect and identify maintaining the classification performance. abnormal driving behaviors. They evaluate through Traditional Bayes Network classifiers have a fixed experiments the performance of NB and support structure that are very difficult to reflect the vector machine. NB has better performance than relationships among nodes (attributes). The authors in support vector machine on detecting and identifying [Xu17] proposed a self-adaptive Bayesian Network various types of the abnormal bus driving behavior classifier based on genetic optimization. Genetic with the accuracy at 98.40%. optimization is used to realize the self-adaptiveness, which means the network structure can be gradually 5 Conclusion optimized when constructing Bayesian Network classifier. Experimental results show that the In this paper we have reviewed recent research work proposed method leads to a high classification on Bayesian Networks. Over the last decade, accuracy than traditional classifier on some Bayesian Networks have become an increasingly benchmarks. popular Artificial Intelligence approach. We have Authors in [Ans17] proposed a framework to reviewed a pool of most recent works done classifying detect the hypervisor attacks in virtual machines using these based on a scheme that consists of three main Bayesian classifier on the publicly available dataset. Bayesian Networks topics: Bayesian Networks They have characterized vulnerabilities of two Structure Learning, Advanced Application of Hypervisors XEN and VMware, based on real-time Bayesian Networks and Bayesian Network attacks. Three attributes namely authentication, Classifiers. We have found that these fields are being integrity impact and confidentiality impact were deeply investigated and interesting approaches are considered for the input feature vector. Experimental being proposed in the field leading also to open results show the parameters of the used attributes that directions for further potential research. have more density for being classified as a hypervisor attack. References In [Kan17] it is proposed a model using a Bayesian classifier for airborne point cloud classification fusing [Kos12] T. J.T. Koski, J. Noble. A Review of multiple data types. The authors based on the analysis Bayesian Networks and Structure Learning. of the characteristics of LiDAR dataset point clouds Journals of the Polish Mathematical Society, and aerial images, they extract the geometric features 40(1):51-103, 2012. doi: from the point clouds and the spectral features from 10.14708/ma.v40i1.278 the optical images. 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