1. Daphne Koller Bayesian Networks Semantics & Factorization Probabilistic Graphical Models Representation

2. Daphne Koller Grade Course Difficulty Student Intelligence Student SAT Reference Letter P(G,D,I,S,L)

3. Daphne Koller IntelligenceDifficulty Grade Letter SAT

4. Daphne Koller IntelligenceDifficulty Grade Letter SAT 0.3 0.08 0.25 0.4 g 2 (B) 0.020.9i 1,d 0 0.70.05i 0,d 1 0.5 0.3 g 1 (A)g 3 (C) 0.2i 1,d 1 0.3i 0,d 0 l1l1 l0l0 0.99 0.4 0.10.9g1g1 0.01g3g3 0.6g2g2 0.2 0.95 s0s0 s1s1 0.8i1i1 0.05i0i0 0.40.6 d1d1 d0d0 0.30.7 i1i1 i0i0

5. Daphne Koller IntelligenceDifficulty Grade Letter SAT P(D)P(I) P(G|I,D)P(S|I) P(L|G) Chain Rule for Bayesian Networks P(D,I,G,S,L) = P(D) P(I) P(G|I,D) P(S|I) P(L|G) Distribution defined as a product of factors!

6. Daphne Koller IntelligenceDifficulty Grade Letter SAT 0.3 0.08 0.25 0.4 g2g2 0.020.9i 1,d 0 0.70.05i 0,d 1 0.5 0.3 g1g1 g3g3 0.2i 1,d 1 0.3i 0,d 0 l1l1 l0l0 0.99 0.4 0.10.9g1g1 0.01g3g3 0.6g2g2 0.2 0.95 s0s0 s1s1 0.8i1i1 0.05i0i0 0.40.6 d1d1 d0d0 0.30.7 i1i1 i0i0 P(d 0, i 1, g 3, s 1, l 1 ) =

7. Template vertLeftWhite2 Defining a joint distribution What is the joint distribution P(D,I,G,S,L)? P(D) P(I) P(G) P(S) P(L) P(D) P(I) P(G|I,D) P(S|I) P(L|G) P(D) P(I) P(G|I) P(G|D) P(S|I) P(L|G) IntelligenceDifficulty Grade Letter SAT P(D|G) P(I|D) P(S|I) P(G|L,I,D) P(L|G)

8. Daphne Koller Bayesian Network A Bayesian network is: – A directed acyclic graph (DAG) G whose nodes represent the random variables X 1,…,X n – For each node X i a CPD P(X i | Par G (X i )) The BN represents a joint distribution via the chain rule for Bayesian networks P(X 1,…,X n ) =  i P(X i | Par G (X i ))

9. Daphne Koller BN Is a Legal Distribution: P ≥ 0

10. Daphne Koller BN Is a Legal Distribution: ∑ P = 1 ∑ D,I,G,S,L P(D,I,G,S,L) = ∑ D,I,G,S,L P(D) P(I) P(G|I,D) P(S|I) P(L|G) = ∑ D,I,G,S P(D) P(I) P(G|I,D) P(S|I) ∑ L P(L|G) = ∑ D,I,G,S P(D) P(I) P(G|I,D) P(S|I) = ∑ D,I,G P(D) P(I) P(G|I,D) ∑ S P(S|I) = ∑ D,I P(D) P(I) ∑ G P(G|I,D)

11. Template vertLeft1 What is the value of 1 P(L) P(G) None of the above

12. Daphne Koller P Factorizes over G Let G be a graph over X 1,…,X n. P factorizes over G if P(X 1,…,X n ) =  i P(X i | Par G (X i ))

13. Daphne Koller Genetic Inheritance Homer Bart Marge LisaMaggie Clancy Jackie Selma Genotype Phenotype AA, AB, AO, BO, BB, OO A, B, AB, O

14. Daphne Koller BNs for Genetic Inheritance G Homer G Bart G Marge G Lisa G Maggie G Clancy G Jackie G Selma B Clancy B Jackie B Selma B Homer B Marge B Bart B Lisa B Maggie