Bayesian Networks

Representation Probabilistic Graphical Models Independencies BayesianNetworks

Independence & Factorization • Factorization of a distribution P over a graph implies independencies in P P(X,Y,Z) = P(X) P(Y) X,Y independent • P(X,Y,Z)  1(X,Y) 2(Y,Z) (X  Y | Z)

d-separation in BNs Definition: X and Y are d-separated in G given Z if there is no active trail in G between X and Y given Z

d-separation examples D I G S L Any node is separated from its non-descendants givens its parents

I-maps • If P satisfies I(H), we say that H is an I-map (independency map) of P I(G) = {(X  Y | Z) : d-sepG(X, Y | Z)}

I-maps P2 P1 G1 G2 D I D I

Factorization  Independence: BNs Theorem: If P factorizes over G, then G is an I-map for P D I G S L

Independence  Factorization Theorem: If G is an I-map for P, then P factorizes over G D I G S L

Factorization  Independence: BNs • Theorem: If P factorizes over G, then G is an I-map for P D I G S L

Summary Two equivalent views of graph structure: • Factorization: G allows P to be represented • I-map: Independencies encoded by G hold in P If P factorizes over a graph G, we can read from the graph independencies that must hold in P (an independency map)

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Bayesian Networks