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Unformatted text preview: ISyE8843A, Brani Vidakovic Handout 17 1 Bayesian Networks Bayesian Networks are directed acyclic graphs (DAG) where the nodes represent random variables and directed edges capture their dependence. Consider the simplest graph A B Figure 1: Simplest A→ B graph. A causes B or B is a consequence of A . We would say that A is a parent of B , B is a child of A , that A influences, or causes B , B depends on A . Also, P ( A,B ) = P ( A ) P ( B  A ) . The independence of two nodes in a DAG depends on their relative position in the graph as well as on the knowledge of other nodes in the graph. The following simple example illustrates the influence of conditioning on the independence. Example 1. Let X 1 and X 2 be results of flips of two fair coins, and X 3 an indicator if the values X 1 and X 2 coincide. Thus, P ( X 1 = H ) = P ( X 1 = T ) = 0 . 5 and P ( X 2 = H ) = P ( X 2 = T ) = 0 . 5 . X 1→ X 3 ← X 2 We are interested in P ( X 3 = ’yes’  X 1 ,X 2 ) . The nodes X 1 and X 2 are marginally independent (when we do not have evidence on X 3 ) but become dependent if the value of X 3 is known, 1 2 = P ( X 1 = T,X 2 = T  X 3 = 1) 6 = P ( X 1 = T  X 3 = 1) P ( X 2 = T  X 3 = 1) = 1 2 · 1 2 . Hard evidence for a node X is evidence that the state of X is takes a particular value. The notion of dseparation 1. In a serial connection from X 1 to X 3 via X 2 , evidence from X 1 to X 3 is blocked only when we have hard evidence about X 2 . X 1 X 2 X 3 Figure 2: Case 1. 2. In a diverging connection where X 1 and X 3 have the common parent X 2 evidence from X 1 to X 3 is blocked only when we have hard evidence about X 2 . 3. In a converging connection where X 3 has parents X 1 and X 2 any evidence about X 3 results in evidence transmitted between X 1 and X 2 . 1 X 1 X 2 X 3 Figure 3: Case 2. X 1 X 3 X 2 Figure 4: Case 3. In Cases 1 and 2 we say that the nodes X 1 and X 3 are dseparated when there is hard evidence about X 2 . In Case 3 , X 1 and X 2 are only dseparated when there is no evidence about X 3 . In general two nodes which are not dseparated are said to be dconnected. These three cases enable us to determine in general whether any two nodes in a given BN are dependent ( dconnected) given the evidence entered in the BN. Formally: Definition of dseparation: Two nodes X and Y in a BN are dseparated if, for all paths between X and Y, there is an intermediate node A for which either: 1. the connection is serial or diverging and the state of A is known for certain; or 2. the connection is diverging and neither A (nor any of its descendants) have received any evidence at all. The problem of exact probabilistic inference in an arbitrary Bayes network is NPHard.[Cooper 1988] NPHard problems are at least as computational complex as NPcomplete problems No algorithms has ever been found which can solve a NPcomplete problem in polynomial time Although it has never been proved whether P = NP or not, many believe that it indeed is not possible. Accordingly, it is unlikely that we couldwhether P = NP or not, many believe that it indeed is not possible....
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This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Tech.
 Spring '11
 VIDAKOVIC

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