The following proofs follow the good proof technique summarized here.

Justification Sources

  • definition 2.3: definition of conditional independence for events (Koller and Friedman, p. 24).
  • definition 2.4: definition of conditional independence for random variables (Koller and Friedman, p. 24).
  • equation 2.8: the “Decomposition” property of conditional independence (Koller and Friedman, p. 25).

Weak Union Property of Conditional Independence

Theorem: “the Weak Union” property of conditional independence: $(X \perp (Y,W) | Z) \Rightarrow (X \perp Y | Z,W)$ (see also equation 2.9 in Koller and Friedman, p. 25).

Proof

Statement Justification
1. $X \perp (Y,W)|Z$ Assumption
2. $P(X|(Y,W),Z) = P(X|Z)$ Step 1, definition 2.3 & definition 2.4 (definitions of conditional independence for events and random variables respectively)
3. $P(X|Y,W,Z) = P(X|Z)$ Ungrouping random variables
4. $X \perp Y | Z $ and $X \perp W|Z$ Step 1, equation 2.8 (decomposition property of conditional independence)
5. $X \perp W | Z$ Step 4, definition of conjuction
6. $P(X|W,Z) = P(X|Z)$ Step 5, definition 2.3
7. $P(X|Y,W,Z) = P(X|W,Z)$ Step 2, Step 6, the transitive property of equality
8. $P(X|Y,(W,Z)) = P(X|(W,Z))$ Step 7, Grouping random variables into sets of random variables
9. $X \perp Y | Z, W$ Step 8, definition 2.3 & definition 2.4
cs-401r/example_proofs.txt · Last modified: 2014/10/23 20:18 by ringger
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