The following proofs follow the good proof technique summarized [[proofs|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 |