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Preliminaries

Probabilistic Graphical Models. Independencies. Preliminaries. Independence. For events , , P    if: P(, ) = P() P() P(|) = P() P(|) = P() 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). Independence. D. I.

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Preliminaries

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  1. Probabilistic Graphical Models Independencies Preliminaries

  2. Independence • For events , , P    if: • P(, ) = P() P() • P(|) = P() • P(|) = P() • 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)

  3. Independence D I P(I,D) G

  4. Conditional Independence • For (sets of) random variables X,Y,Z P (X  Y | Z) if: • P(X, Y|Z) = P(X|Z) P(Y|Z) • P(X|Y,Z) = P(X|Z) • P(Y|X,Z) = P(X|Z) • P(X,Y,Z)  1(X,Y) 2(Y,Z)

  5. Conditional Independence Coin X1 X2

  6. Conditional Independence I P(S,G | i0) G S

  7. Conditioning can Lose Independences P(I,D | g1)

  8. END END END

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