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

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Presentation transcript:

Daphne Koller Bayesian Networks Semantics & Factorization Probabilistic Graphical Models Representation

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

Daphne Koller IntelligenceDifficulty Grade Letter SAT

Daphne Koller IntelligenceDifficulty Grade Letter SAT g 2 (B) i 1,d i 0,d g 1 (A)g 3 (C) 0.2i 1,d 1 0.3i 0,d 0 l1l1 l0l g1g1 0.01g3g3 0.6g2g s0s0 s1s1 0.8i1i1 0.05i0i d1d1 d0d i1i1 i0i0

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!

Daphne Koller IntelligenceDifficulty Grade Letter SAT g2g i 1,d i 0,d g1g1 g3g3 0.2i 1,d 1 0.3i 0,d 0 l1l1 l0l g1g1 0.01g3g3 0.6g2g s0s0 s1s1 0.8i1i1 0.05i0i d1d1 d0d i1i1 i0i0 P(d 0, i 1, g 3, s 1, l 1 ) =

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)

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 ))

Daphne Koller BN Is a Legal Distribution: P ≥ 0

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)

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

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 ))

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

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