bayesian networks
Post on 03-Jan-2016
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Daphne Koller
Independencies
BayesianNetworks
ProbabilisticGraphicalModels
Representation
Daphne Koller
Independence & Factorization
• Factorization of a distribution P over a graph implies independencies in P
P(X,Y,Z) 1(X,Y) 2(Y,Z)
P(X,Y,Z) = P(X) P(Y) X,Y independent
(X Y | Z)
Daphne Koller
d-separation in BNsDefinition: X and Y are d-separated in G given
Z if there is no active trail in G between X and Y given Z
Daphne Koller
d-separation examplesID
G
L
S
Any node is separated from its non-descendants givens its parents
Daphne Koller
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)}
Daphne Koller
I-maps
I D Prob
i0 d0 0.42
i0 d1 0.18
i1 d0 0.28
i1 d1 0.12
I D Prob.i0 d0 0.282i0 d1 0.02 i1 d0 0.564i1 d1 0.134
G1
P2
IDID
P1
G2
Daphne Koller
Factorization Independence: BNs
Theorem: If P factorizes over G, then G is an I-map for P
ID
G
L
S
Daphne Koller
Independence Factorization
Theorem: If G is an I-map for P, then P factorizes over G
ID
G
L
S
Daphne Koller
Factorization Independence: BNs
• Theorem: If P factorizes over G, then G is an I-map for P
ID
G
L
S
Daphne Koller
SummaryTwo 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)
Daphne Koller
END END END
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