Independence
- For events
α,β ,P⊨α⊥β if:
P(α,β)=P(α)P(β) P(α|β)=P(α) P(β|α)=P(β)
- For random variables
X,Y ,P⊨X⊥Y if:
P(X,Y)=P(X)P(Y) P(X|Y)=P(X) P(Y|X)=P(Y)
Conditional Independence
For (sets of) random variables
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d-seperated
Factorization ⇒ Independence: BNs
- Theorem: If
P factorized overG , andd-sepG(X,Y|Z) , thenP satisfies(X⊥Y|Z) - Any node is d-seperated from its non-descendants given its parents.
- If
P factorizes overG , then inP , any variable is independent of its non-descendants given its parents. - I-maps
- d-separation in
G⇒P satisfies corresponding independence statement
I(G)={(X,Y|Z):d-sepG(X,Y|Z)} - Definition: If
P satisfiesI(G) , we say thatG is an I-map (independency map) ofP .
- d-separation in
- Theorem: If
P factorized overG , thenG is an I-map ofP .
Independence ⇒ Factorization
- Theorem: If
G is an I-map forP , thenP factorized overG .
Summary
- Two equivalent views of graph structure
- Factorization:
G allowsP to be represented - I-map: Independencies encoded in
G hold inP
- Factorization:
- If
P factorizes over a graphG , we can read from the graph independencies that must hold inP (an independency map)