Lecture12 - Bayesian networks Chapter 14 Section 1 2...

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Bayesian networks Chapter 14 Section 1 – 2
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Bayesian networks A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions Syntax: a set of nodes, one per variable a directed, acyclic graph (link ≈ "directly influences") a conditional distribution for each node given its parents: P (X i | Parents (X i ))
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Example Topology of network encodes conditional independence assertions: Weather is independent of the other variables Toothache and Catch are conditionally independent given Cavity
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Example I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar? Variables: Burglary , Earthquake , Alarm , JohnCalls , MaryCalls Network topology reflects "causal" knowledge: A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call
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Example contd.
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Compactness If each variable has no more than k parents, the complete network requires O(n · 2 k ) numbers I.e., grows linearly with n , vs. O(2 n ) for the full joint distribution For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 2 5 - 1 = 31)
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Semantics
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Constructing Bayesian networks 1. Choose an ordering of variables X 1 , … , X n 2. For i = 1 to n add X i to the network select parents from X 1 , … ,X i-1 such that P (X i | Parents(X i )) = P (X i | X 1 , . .. X i-1 ) This choice of parents guarantees: P (X 1 , … ,X n ) = π i =1 P (X i | X 1 , … , X i-1 ) (chain rule) = π i =1 P (X i | Parents(X i )) (by construction) n n
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Suppose we choose the ordering M, J, A, B, E P (J | M) = P (J)? Example
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Lecture12 - Bayesian networks Chapter 14 Section 1 2...

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