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