Data Mining and Knowledge Discovery Handbook, 2 Edition part 20 ppt

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Data Mining and Knowledge Discovery Handbook, 2 Edition part 20 ppt

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L., Finding Small Equivalent Decision Trees is Hard, International Journal of Foundations of Computer Science, 11(2): 343-354, 2000. 10 Bayesian Networks Paola Sebastiani 1 , Maria M. Abad 2 , and Marco F. Ramoni 3 1 Department of Biostatistics Boston University sebas@bu.edu 2 Software Engineering Department University of Granada, Spain mabad@ugr.es 3 Departments of Pediatrics and Medicine Harvard University marco ramoni@harvard.edu Summary. Bayesian networks are today one of the most promising approaches to Data Min- ing and knowledge discovery in databases. This chapter reviews the fundamental aspects of Bayesian networks and some of their technical aspects, with a particular emphasis on the methods to induce Bayesian networks from different types of data. Basic notions are illus- trated through the detailed descriptions of two Bayesian network applications: one to survey data and one to marketing data. Key words: Bayesian networks, probabilistic graphical models, machine learning, statistics. 10.1 Introduction Born at the intersection of Artificial Intelligence, statistics and probability, Bayesian networks (Pearl, 1988) are a representation formalism at the cutting edge of knowl- edge discovery and Data Mining (Heckerman, 1997, Madigan and Ridgeway, 2003, Madigan and York, 1995). Bayesian networks belong to a more general class of mod- els called probabilistic graphical models (Whittaker, 1990,Lauritzen, 1996) that arise from the combination of graph theory and probability theory and their success rests on their ability to handle complex probabilistic models by decomposing them into smaller, amenable components. A probabilistic graphical model is defined by a graph where nodes represent stochastic variables and arcs represent dependencies among such variables. These arcs are annotated by probability distribution shaping the in- teraction between the linked variables. A probabilistic graphical model is called a Bayesian network when the graph connecting its variables is a directed acyclic graph (DAG). This graph represents conditional independence assumptions that are used to factorize the joint probability distribution of the network variables thus making the process of learning from large database amenable to computations. A Bayesian network induced from data can be used to investigate distant relationships between O. Maimon, L. Rokach (eds.), Data Mining and Knowledge Discovery Handbook, 2nd ed., DOI 10.1007/978-0-387-09823-4_10, © Springer Science+Business Media, LLC 2010 176 Paola Sebastiani, Maria M. Abad, and Marco F. Ramoni variables, as well as making prediction and explanation, by computing the condi- tional probability distribution of one variable, given the values of some others. The origins of Bayesian networks can be traced back as far as the early decades of the 20th century, when Sewell Wright developed path analysis to aid the study of genetic inheritance (Wright, 1923,Wright, 1934). In their current form, Bayesian net- works were introduced in the early 80s as a knowledge representation formalism to encode and use the information acquired from human experts in automated reasoning systems to perform diagnostic, predictive, and explanatory tasks (Pearl, 1988, Char- niak, 1991). Their intuitive graphical nature and their principled probabilistic foun- dations were very attractive features to acquire and represent information burdened by uncertainty. The development of amenable algorithms to propagate probabilistic information through the graph (Lauritzen and Spiegelhalter, 1988, Pearl, 1988) put Bayesian networks at the forefront of Artificial Intelligence research. Around same time, the machine learning community came to the realization that the sound prob- abilistic nature of Bayesian networks provided straightforward ways to learn them from data. As Bayesian networks encode assumptions of conditional independence, the first machine learning approaches to Bayesian networks consisted of searching for conditional independence structures in the data and encoding them as a Bayesian network (Glymour et al., 1987, Pearl, 1988). Shortly thereafter, Cooper and Her- skovitz (Cooper and Herskovitz, 1992) introduced a Bayesian method, further re- fined by (Heckerman et al., 1995), to learn Bayesian networks from data. These re- sults spurred the interest of the Data Mining and knowledge discovery community in the unique features of Bayesian networks (Heckerman, 1997): a highly symbolic for- malism, originally developed to be used and understood by humans, well-grounded on the sound foundations of statistics and probability theory, able to capture complex interaction mechanisms and to perform prediction and classification. 10.2 Representation A Bayesian network has two components: a directed acyclic graph and a probability distribution. Nodes in the directed acyclic graph represent stochastic variables and arcs represent directed dependencies among variables that are quantified by condi- tional probability distributions. As an example, consider the simple scenario in which two variables control the value of a third. We denote the three variables with the letters A, B and C, and we assume that each is bearing two states: “True” and “False”. The Bayesian network in Figure 10.1 describes the dependency of the three variables with a directed acyclic graph, in which the two arcs pointing to the node C represent the joint action of the two variables A and B. Also, the absence of any directed arc between A and B describes the marginal independence of the two variables that become dependent when we condition on the phenotype. Following the direction of the arrows, we call the node C a child of A and B, which become its parents. The Bayesian network in Figure 10.1 let us decompose the overall joint probability distribution of the three variables that would consist of 2 3 −1 = 7 parameters into three probability distri- 10 Bayesian Networks 177 Fig. 10.1. A network describing the impact of two variables (nodes A and B) on a third one (node C). Each node in the network is associated with a probability table that describes the conditional distribution of the node, given its parents. butions, one conditional distribution for the variable C given the parents, and two marginal distributions for the two parent variables A and B. These probabilities are specified by 1 +1 + 4 = 6 parameters. The decomposition is one of the key factors to provide both a verbal and a human understandable description of the system and to efficiently store and handle this distribution, which grows exponentially with the number of variables in the domain. The second key factor is the use of conditional independence between the network variables to break down their overall distribution into connected modules. Suppose we have three random variables Y 1 ,Y 2 ,Y 3 . Then Y 1 and Y 2 are indepen- dent given Y 3 if the conditional distribution of Y 1 ,givenY 2 ,Y 3 is only a function of Y 3 . Formally: p(y 1 |y 2 ,y 3 )=p(y 1 |y 3 ) where p(y|x) denotes the conditional probability/density of Y ,givenX = x. We use capital letters to denote random variables, and small letters to denote their values. We also use the notation Y 1 ⊥Y 2 |Y 3 to denote the conditional independence of Y 1 and Y 2 given Y 3 . Conditional and marginal independence are substantially different concepts. For example two variables can be marginally independent, but they may be dependent when we condition on a third variable. The directed acyclic graph in Figure 10.1 shows this property: the two parent variables are marginally independent, but they 178 Paola Sebastiani, Maria M. Abad, and Marco F. Ramoni become dependent when we condition on their common child. A well known con- sequence of this fact is the Simpson’s paradox (Whittaker, 1990) : two variables are independent but once a shared child variable is observed they become dependent. Fig. 10.2. A network encoding the conditional independence of Y 1 ,Y 2 given the common par- ent Y 3 . The panel in the middle shows that the distribution of Y 2 changes with Y 1 and hence the two variables are conditionally dependent. Conversely, two variables that are marginally dependent may be made condi- tionally independent by introducing a third variable. This situation is represented by the directed acyclic graph in Figure 10.2, which shows two children nodes (Y 1 and Y 2 ) with a common parent Y 3 . In this case, the two children nodes are independent, given the common parent, but they may become dependent when we marginalize the common parent out. The overall list of marginal and conditional independencies represented by the di- rected acyclic graph is summarized by the local and global Markov properties (Lau- ritzen, 1996) that are exemplified in Figure 10.3 using a network of seven variables. The local Markov property states that each node is independent of its non descendant given the parent nodes and leads to a direct factorization of the joint distribution of the network variables into the product of the conditional distribution of each vari- able Y i given its parents Pa(y i ). Therefore, the joint probability (or density) of the v network variables can be written as: p(y 1 , ,y v )= ∏ i p(y i |pa(y i )). (10.1) In this equation, pa(y i ) denotes a set of values of Pa(Y i ). This property is the core of many search algorithms for learning Bayesian networks from data. With this de- 10 Bayesian Networks 179 Fig. 10.3. A Bayesian network with seven variables and some of the Markov properties repre- sented by its directed acyclic graph. The panel on the left describes the local Markov property encoded by a directed acyclic graph and lists the three Markov properties that are represented by the graph in the middle. The panel on the right describes the global Markov property and lists three of the seven global Markov properties represented by the graph in the middle. The vector in bold denotes the set of variables represented by the nodes in the graph. composition, the overall distribution is broken into modules that can be interrelated, and the network summarizes all significant dependencies without information disin- tegration. Suppose, for example, the variable in the network in Figure 10.3 are all categorical. Then the joint probability p(y 1 , ,y 7 ) can be written as the product of seven conditional distributions: p(y 1 )p(y 2 )p(y 3 |y 1 ,y 2 )p(y 4 )p(y 5 |y 3 )p(y 6 |y 3 ,y 4 )p(y 7 |y 5 ,y 6 ). The global Markov property, on the other hand, summarizes all conditional indepen- dencies embedded by the directed acyclic graph by identifying the Markov Blanket of each node (Figure 10.3). 10.3 Reasoning The modularity induced by the Markov properties encoded by the directed acyclic graph is the core of many search algorithms for learning Bayesian networks from data. By the Markov properties, the overall distribution is broken into modules that can be interrelated, and the network summarizes all significant dependencies with- out information disintegration. In the network in Figure 10.3, for example, we can compute the probability distribution of the variable Y 7 , given that the variable Y 1 is observed to take a particular value (prediction) or, vice versa, we can compute the conditional distribution of Y 1 given the values of some other variables in the network (explanation). In this way, a Bayesian network becomes a complete simulation sys- tem able to forecast the value of unobserved variables under hypothetical conditions and, conversely, able to find the most probable set of initial conditions leading to observed situation. . Number 2, 20 05b, pp 131–158. Rokach, L. and Maimon, O., Clustering methods, Data Mining and Knowledge Discovery Handbook, pp. 321 –3 52, 20 05, Springer. Rokach, L. and Maimon, O., Data mining for. Tree Construction of Large Datasets ,Data Mining and Knowledge Discovery, 4, 2/ 3) 127 -1 62, 20 00. Gelfand S. B., Ravishankar C. S., and Delp E. J., An iterative growing and pruning algo- rithm for. Information and Knowledge Systems, Lecture Notes in Computer Science, Springer, pp. 178-196, 20 02. Maimon, O. and Rokach, L., Decomposition Methodology for Knowledge Discovery and Data Mining: Theory and

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