I'm reading Murphy's Machine Learning: A probabilistic perspective. Exercise 2.9 asks:
Exercise 2.9
Conditional independence
Are the following properties true? Prove or disprove. Note that we are not restricting attention to distributions that can be represented by a graphical model.
- True or false? $$ (X \perp W \mid Z, Y)\wedge (X \perp Y \mid Z) \implies (X \perp Y, W \mid Z) $$
where $(A \perp B \mid C)$ means $$ p(A, B \mid C) = p(A \mid C) p(B \mid C) $$
I need clarification on how to parse $$ (X \perp Y, W \mid Z) $$
Does this mean that $$ p(X, Y, W \mid Z) = p(X \mid Z)p(Y, W \mid Z) $$