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M. Julia Flores (University of Castilla – La Mancha (UCLM), Instituto de Investigación en informática de Albacete, Spain), José A. Gámez (University of Castilla – La Mancha (UCLM), Instituto de Investigación en informática de Albacete, Spain) and Ana M. Martínez (University of Castilla – La Mancha (UCLM), Instituto de Investigación en informática de Albacete, Spain)

Copyright: © 2012
|Pages: 31

DOI: 10.4018/978-1-4666-1806-0.ch005

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TopThe task of classification is one of the most popular and, therefore, important tasks in data mining, as it is applied in many real applications. In this context, the basic *classification* task involves learning a model (or generalization) from a set of labelled data, in order to assign one label to every new example. The model learning phase can be more or less complex, to such a degree that most of the work might be carried out in the assignation phase (as in *lazy classifiers*) or, often simply called, classification phase. As world is not deterministic, we will have to manage with uncertainty in classification.

Formally, a model is learnt from a dataset with *t* examples and *n* attributes, all of them with known labels given by a special attribute called *class*, *C*. Hence it is also often referred to as *supervised classification* in contrast to *unsupervised classification* or *clustering*, where the labels are not known a priori. For every example of the type where each *a _{i}* is the value for the attribute

There exist multiple paradigms for classification, such as Bayesian networks, decision trees, rule induction, artificial neural networks, genetic programming, support vector machines, etc. In this chapter we focus on the first of these, which might be seen as a combination of statistical techniques and graphical models. BNs provide several advantages to the classification task:

*•*The networks store information about dependencies existing among the variables involved, which makes them capable of inherently dealing with uncertainty, very frequently present in real world.

*•*The graphical representation through the Bayesian network facilitates the interpretation and formulation of conclusions about the domain of study.

*•*In addition, Bayesian network classifiers can combine causal relationships with probabilistic logic, which helps to incorporate expert knowledge into the model.

Thus, one of the greatest advantages of the BNs is that they can represent both the qualitative and the quantitative aspect of the problem. The former is encoded in a directed acyclic graph (DAG), whereas the latter involves storing a probability table for every node conditioned on its parents. Even though the conditional probability distribution can be represented in several ways, the most common representation is the use of tables, i.e. conditional probability tables (CPT).

In a DAG, each node represents a variable; an arc represents a direct dependence between the pair of nodes connected. If there is a directed arc from *X* to *Y*, it means that *X* is the *parent* of *Y* and *Y* is *child* of *X*. Furthermore, if there exists a directed path from *X* to *Z*, it implies that *X* is an *ancestor* of *Z*, while *Z* is a *descendant* of *X*.

Through the property of *conditional independence*, which states that a node is conditionally independent of its non-descendant given its parents, we can represent the joint probability distribution of a Bayesian network by the product of the CPTs associated with each of its nodes.

In classification, we want to obtain i.e. the conditional probability for *C* given The accurate estimation of the probabilities a posteriori for every combination of the class labels and the values of the attributes is unfeasible in practice, as it requires a large amount of training data even with a moderate number of attributes. That is why it is convenient to resort to the Bayes theorem:

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