handout17

handout17 - ISyE8843A Brani Vidakovic 1 Handout 17 Bayesian...

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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 d -separation 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

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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 d -separated when there is hard evidence about X 2 . In Case 3 , X 1 and X 2 are only d -separated when there is no evidence about X 3 . In general two nodes which are not d -separated are said to be d -connected. These three cases enable us to determine in general whether any two nodes in a given BN are dependent ( d -connected) given the evidence entered in the BN. Formally: Definition of d -separation: Two nodes X and Y in a BN are d -separated 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 NP-Hard.[Cooper 1988] NP-Hard problems are at least as computational complex as NP-complete problems No algorithms has ever been found which can solve a NP-complete 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 could develop an general-purpose efficient exact method for propagating probabilities in an arbitrary network
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