1. Elimination in Chains A B C E D

2. Elimination in Chains
By the chain rule for probabilities: A B C E D

3. Elimination in Chains
Rearranging terms ... A B C E D

4. Elimination in Chains
Perform the innermost summation X A B C E D

5. Elimination in Chains
Rearrange and then sum again X X A B C E D

6. a1 b1 c1
0.5·0.5 = 0.25 c2
0.5·0.7 = 0.35 b2
0.8·0.1 = 0.08
0.8·0.2 = 0.16 a2
0.1·0.5 = 0.05
0.1·0.7 = 0.07
0·0.1 = 0
0·0.2 = 0 a3
0.3·0.5 = 0.15
0.3·0.7 = 0.21
0.9·0.1 = 0.09
0.9·0.2 = 0.18 a1 b1
0.5 b2
0.8 a2
0.1 a3
0.3
0.9 b1 c1
0.5 c2
0.7 b2
0.1
0.2

7. a1 b1 c1
0.25 c2
0.35 b2
0.08
0.16 a2
0.05
0.07 a3
0.15
0.21
0.09
0.18 a1 c1
0.33 c2
0.51 a2
0.05
0.07 a3
0.24
0.39

8. A More Complex Example “Asia” network: Visit to Asia Smoking
Lung Cancer
Tuberculosis
Abnormality
in Chest
Bronchitis
X-Ray
Dyspnea

9. Need to eliminate: V,S,X,T,L,A,B Initial factors
We want to compute P(D)
Need to eliminate: V,S,X,T,L,A,B
Initial factors

10. Need to eliminate: V,S,X,T,L,A,B Initial factors
We want to compute P(D)
Need to eliminate: V,S,X,T,L,A,B
Initial factors

11. Need to eliminate: V,S,X,T,L,A,B Current factors
We want to compute P(D)
Need to eliminate: V,S,X,T,L,A,B
Current factors
Eliminate: V
Compute:
Note: τ1(T) = P(T)

12. Need to eliminate: S,X,T,L,A,B Current factorsV S L T A B X D
We want to compute P(D)
Need to eliminate: S,X,T,L,A,B
Current factors
Eliminate: S
Compute:

13. Need to eliminate: X,T,L,A,B Current factorsV S L T A B X D
We want to compute P(D)
Need to eliminate: X,T,L,A,B
Current factors
Eliminate: X
Compute:
Note: τ3(A) = 1 for all values of A !!

14. Need to eliminate: T,L,A,B Current factorsV S L T A B X D
We want to compute P(D)
Need to eliminate: T,L,A,B
Current factors
Eliminate: T
Compute:

15. We want to compute P(D) Need to eliminate: L,A,B Current factorsV S L T A B X D
We want to compute P(D)
Need to eliminate: L,A,B
Current factors
Eliminate: L
Compute:

16. We want to compute P(D) Need to eliminate: A,B Current factorsV S L T A B X D
We want to compute P(D)
Need to eliminate: A,B
Current factors
Eliminate: A
Compute:

17. We want to compute P(D) Need to eliminate: B Current factorsV S L T A B X D
We want to compute P(D)
Need to eliminate: B
Current factors
Eliminate: B
Compute:
Note: τ7(D) is P(D)

18. Dealing with evidence How do we deal with evidence?S L T A B X D
How do we deal with evidence?
Suppose get evidence V = v, S = s, D = d
We want to compute P(L, V = v, S = s, D = d)

19. 0.25 c1 b1 a1
0.35 c2
0.16 b2
0.08
0.05 a2
0.07
0.18 a3
0.09
0.21
0.15

20. 0.25 c1 b1 a1
0.35 c2
0.16 b2
0.08
0.05 a2
0.07
0.18 a3
0.09
0.21
0.15 a1 b1 c1
0.25 b2
0.08 a2
0.05 a3
0.15
0.09

21. Dealing with Evidence Compute P(L, V = v, S = s, D = d )B X D
Compute P(L, V = v, S = s, D = d )
Initial factors, after setting evidence:

22. Example Want to compute P(L) Moralizing V S L T A B X D V S D T L A B

23. Example Want to compute P(L) Moralizing Eliminating V V S L T A B X D

24. Example Want to compute P(L) Moralizing Eliminating V Eliminating S VB X D
Want to compute P(L)
Moralizing
Eliminating V
Eliminating S V S T L A B X D

25. Example Want to compute P(L) Moralizing Eliminating V Eliminating SB X D
Want to compute P(L)
Moralizing
Eliminating V
Eliminating S
Eliminating X V S T L A B X D

26. Example Want to compute P(D) Moralizing Eliminating V Eliminating SB X D
Want to compute P(D)
Moralizing
Eliminating V
Eliminating S
Eliminating X
Eliminating T V S T L A B X D

27. Example Want to compute P(D) Moralizing Eliminating V Eliminating SB X D
Want to compute P(D)
Moralizing
Eliminating V
Eliminating S
Eliminating X
Eliminating T
Eliminating L V S T L A B X D

28. Example Want to compute P(D) Moralizing Eliminating V Eliminating SB X D
Want to compute P(D)
Moralizing
Eliminating V
Eliminating S
Eliminating X
Eliminating T
Eliminating L
Eliminating A, B V S T L A B X D

29. L T A B X V S D