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Review of Bayes’ Rule

BIOL 301 Guest Lecture: Reasoning Under Uncertainty (Intro to Bayes Networks) Simon D. Levy CSCI Department 8 April 2010. Rev. Thomas Bayes (1702-1761). Review of Bayes’ Rule. From the Product Rule:. P( A | B ) = P( A & B ) / P(B). P( A & B ) = P( A | B ) * P( B ).

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Review of Bayes’ Rule

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  1. BIOL 301 Guest Lecture: Reasoning Under Uncertainty(Intro to Bayes Networks)Simon D. LevyCSCI Department8 April 2010

  2. Rev. Thomas Bayes (1702-1761) Review of Bayes’ Rule • From the Product Rule: • P(A|B) = P(A &B) / P(B) • P(A &B) = P(A|B) * P(B) = P(B|A) * P(A) • We derive Bayes’ Rule by substitution: • P(A|B) = P(A &B) / P(B) • = P(B|A) * P(A) / P(B)

  3. Real-World Problems May Involve Many Variables http://www.bayesia.com/assets/images/content/produits/blab/tutoriel/en/chapitre-3/image026.jpg

  4. Real-World Problems May Involve Many Variables

  5. Variables Are Typically Oberserved Simultaneously (Confounded) FeverAcheVirusP No No No .950 No No Yes .002 No Yes No .032 No Yes Yes .002 Yes No No .002 Yes No Yes .001 Yes Yes No .010 Yes Yes Yes .001 So how do we compute P(V=Yes), P(F=Yes & A=No), etc.?

  6. Marginalization P(V=Yes): FeverAcheVirusP No No No .950 No No Yes .002 No Yes No .032 No Yes Yes .002 Yes No No .002 Yes No Yes .001 Yes Yes No .010 Yes Yes Yes .001 ______ Sum = .006

  7. Marginalization P(F=Yes & A=No): FeverAcheVirusP No No No .950 No No Yes .002 No Yes No .032 No Yes Yes .002 Yes No No .002 Yes No Yes .001 Yes Yes No .010 Yes Yes Yes .001 ______ Sum = .003

  8. Combinatorial Explosion (The “Curse of Dimensionality”) Assuming (unrealistically) only two values (Yes/No) per variable: # Variables# of Rows in Table 1 2 2 4 3 8 4 16 5 32 6 64 : : 20 1,048,576

  9. Solution: Local Causality + Belief Propagation

  10. Local Causality

  11. P(B) .001 P(E) .002 B E P(A) T T .95 T F .94 F T .29 F F .001 Recover Joint From Prior & Posterior BEAProb T T T .001*.002*.95 = .000001900 T T F .001*.002*.05 = .000000100 T F T .001*.998*.94 = .000938120 T F F .001*.998*.06 = .000059880 F T T .999*.002*.29 = .000579420 F T F .999*.002*.71 = .001418580 F F T .999*.998*.001 = .000997002 F F F .999*.998*.999 = .996004998

  12. Belief Propagation • Consider just B → A → J • P(J=T | B=T) = P(J=T | A=T) * P(A=T | B=T) • Then use Bayes’ Rule and marginalization to answer more sophisticated queries like • P(B=T | J=F & E=F & M=T)

  13. Multiply-Connected Networks Cloudy Rain Sprinkler Wet Grass

  14. Clustering (“Mega Nodes”) Cloudy Sprinkler Rain Sprinkler Rain Sprinkler Rain Wet Grass

  15. A C G B D E H F Junction Tree Algorithm (Huang & Darwiche 1994)

  16. A A C C G G B B D D E H E H F F Junction Tree Algorithm (Huang & Darwiche 1994) “Moralize””

  17. A C G B D E H F Junction Tree Algorithm (Huang & Darwiche 1994)

  18. A A C C G G B B D D E E H H F F Junction Tree Algorithm (Huang & Darwiche 1994) Triangulate

  19. A C G B D E H F Junction Tree Algorithm (Huang & Darwiche 1994)

  20. ABD ADE ACE CEG AD AE CE A EG DE C G B DEF EGH D E H F Junction Tree Algorithm (Huang & Darwiche 1994)

  21. ABD ADE ACE CEG AD AE CE EG DE DEF EGH “Message-Passing” Observe A=T

  22. ABD ADE ACE CEG AD AE CE EG DE DEF EGH “Message-Passing” Pick a cluster containing A:

  23. ABD ADE ACE CEG AD AE CE EG DE DEF EGH “Message-Passing” Pass messages to propagate evidence:

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