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bayesian networks simple graphical notation
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- 3. Bayesian networks • Asimple, 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 (Xi | Parents (Xi)) • In the simplest case, conditional distribution represented as a conditional probability table (CPT) giving the distribution over Xi for each combination of parent values –
- 4. Example • Topology ofnetwork encodes conditional independence assertions: • Weather is independent of the other variables • Toothache and Catch are conditionally independent given Cavity
- 5. Example • I'm atwork, 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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- 7. Compactness • A CPTfor Boolean Xi with k Boolean parents has 2k rows for the combinations of parent values • Each row requires one number p for Xi = true (the number for Xi = false is just 1-p) • If each variable has no more than k parents, the complete network requires O(n · 2k ) numbers • I.e., grows linearly with n, vs. O(2n ) for the full joint distribution • For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25 -1 = 31)
- 8. Semantics The full jointdistribution is defined as the product of the local conditional distributions: P (X1, … ,Xn) = πi = 1 P (Xi | Parents(Xi)) e.g., P(j m a b e) = P (j | a) P (m | a) P (a | b, e) P (b) P (e) • • n
- 9. Constructing Bayesian networks •1. Choose an ordering of variables X1, … ,Xn • 2. For i = 1 to n – add Xi to the network – select parents from X1, … ,Xi-1 such that P (Xi | Parents(Xi)) = P (Xi | X1, ... Xi-1) This choice of parents guarantees: P (X1, … ,Xn) = πi =1 P (Xi | X1, … , Xi-1) = πi =1P (Xi | Parents(Xi)) (by construction) (chain rule) • – n n
- 10. • Suppose wechoose the ordering M, J, A, B, E P(J | M) = P(J)? • • Example
- 11. • Suppose wechoose the ordering M, J, A, B, E P(J | M) = P(J)? P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? • No • Example
- 12. • Suppose wechoose the ordering M, J, A, B, E P(J | M) = P(J)? P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? P(B | A, J, M) = P(B)? • No • Example
- 13. • Suppose wechoose the ordering M, J, A, B, E P(J | M) = P(J)? P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A ,J, M) = P(E | A)? P(E | B, A, J, M) = P(E | A, B)? • No • Example
- 14. • Suppose wechoose the ordering M, J, A, B, E P(J | M) = P(J)? P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A ,J, M) = P(E | A)? No P(E | B, A, J, M) = P(E | A, B)? Yes • No • Example
- 15. Example contd. • Decidingconditional independence is hard in noncausal directions • (Causal models and conditional independence seem hardwired for humans!) • Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed • • •
- 16. Summary • Bayesian networksprovide a natural representation for (causally induced) conditional independence • Topology + CPTs = compact representation of joint distribution • Generally easy for domain experts to construct