Elimination in Chains A B C E D

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

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

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

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

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

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

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

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

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

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)

Need to eliminate: S,X,T,L,A,B Current factors V 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:

Need to eliminate: X,T,L,A,B Current factors V 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 !!

Need to eliminate: T,L,A,B Current factors V 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:

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

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

We want to compute P(D) Need to eliminate: B Current factors V 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)

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)

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

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

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:

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

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

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

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

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

Example Want to compute P(D) Moralizing Eliminating V Eliminating S B 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

Example Want to compute P(D) Moralizing Eliminating V Eliminating S B 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

L T A B X V S D