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Bayes Nets

Bayes Nets. 10-701 recitation 04-02-2013. Bayes Nets. Allergy. Nose. Sinus. Flu. Headache. Represent dependencies between variables Compact representation of probability distribution. Encodes causal relationships. Conditional independence. N ose. Sinus. Sinus. Flu. Headache. Nose.

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Bayes Nets

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  1. Bayes Nets 10-701 recitation 04-02-2013

  2. Bayes Nets Allergy Nose Sinus Flu Headache • Represent dependencies between variables • Compact representation of probability distribution • Encodes causal relationships

  3. Conditional independence Nose Sinus Sinus Flu Headache Nose • P(X,Y|Z) = P(X|Z) x P(Y|Z) F not⊥N Nnot⊥H F⊥N | S N⊥H | S

  4. Conditional independence Sinus Flu Allergy • Explaining away: • P(F = t | S = t) is high • But P(F = t | S = t , A = t) is lower F⊥A F not⊥A | S

  5. Joint probability distribution • Chain rule of probability: P(X1, X2, …, Xn) = P( X1) P( X2|X1) … P(Xn|X1,X2…Xn-1)

  6. Joint probability distribution Allergy Nose Sinus Flu Headache • Chain rule of probability: P(F,A,S,H,N) = P(F) P(A|F) P(S|A,F) P(H|F,A,S) P(N|F,A,S,H) Table with 25 entries!

  7. Joint probability distribution Allergy Nose Sinus Flu Headache • Local markov assumption: A variable X is independent of it’s non-descendants given it’s parents P(F,A,S,H,N) = P(F) P(A) P(S|A,F) P(H|S) P(N|S) P(F) P(A) P(S|F,A) P(H|S) P(N|S)

  8. Queries, Inference Allergy Nose=t Sinus Flu Headache • P(F = t | N = t) ? • P(F=t|N=t) = P(F=t,N=t)/P(N=t) P(F,N=t) = ΣA,S,H P(F,A,S,H,N=t) = ΣA,S,H P(F) P(A) P(S|A,F) P(H|S) P(N=t|S) P(F) P(A) P(S|F,A) P(H|S) P(N|S)

  9. Moralizing the graph Allergy Nose=t Sinus Flu Headache • Eliminating A will create a factor with F and S • To assess complexity we can moralize the graph: connect parents

  10. Chose an optimal order Allergy Nose=t Sinus Flu If we start with H: P(F,N=t) = ΣA,S P(F) P(A) P(S|A,F) P(N=t |S) ΣH P(H|S) = ΣA,S P(F) P(A) P(S|A,F) P(N=t |S) =1

  11. Allergy Nose=t Sinus Flu Removing S P(F,N=t)= ΣA,S P(F) P(A) P(S|A,F) P(N=t |S) = ΣA P(F) P(A) ΣsP(S|A,F) P(N=t |S) = ΣA P(F) P(A) g1(F,A)

  12. Allergy Nose=t Flu Removing A P(F,N=t)= P(F) ΣA P(A) g1(F,A) = P(F) g2(F) P(F=t|N=t) = P(F=t,N=t)/P(N=t) P(N=t) = ΣFP(F,N=t) =P(N=t|F)

  13. Independencies and active trails Is A⊥H? When is it not? A is not⊥Hwhen given C and F or F’ or F’’ and not {B,D,E,G}

  14. Independencies and active trails • Active trail between variables X1,X2…Xn-1 when: • Xi-1 -> Xi -> Xi+1 and Xi not observed • Xi-1 <- Xi <- Xi+1 and Xi not observed • Xi-1 <- Xi -> Xi+1 and Xi not observed • Xi-1 -> Xi <- Xi+1 and Xi or one of its descendants is observed

  15. Independencies and active trails A⊥B ?

  16. Independencies and active trails B⊥G |E ?

  17. Independencies and active trails I⊥J |K ?

  18. Independencies and active trails E⊥F |K ?

  19. Independencies and active trails F⊥K |I ?

  20. Independencies and active trails E⊥F |I,K ?

  21. Independencies and active trails F⊥G |H ?

  22. Independencies and active trails F⊥G |H ,A ?

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