Help proving $Pr(\mathcal{X})= \phi_1(X,Z)\phi_2(Y,Z)$ if $ P \models (X \perp Y | Z)$ and $\mathcal{X}=X \cup Y \cup Z$

Question

I was trying to prove the following:

if $X,Y,Z$ were three disjoint subsets of variables such that $\mathcal{X}=X \cup Y \cup Z$, Prove that $ P \models (X \perp Y \mid Z)$ if and only if we can write P in the form:

$$P(\mathcal{X})= \phi_1(X,Z)\phi_2(Y,Z)$$

I was a little stuck but was unsure if I it was because I didn't know how to do the proof or if I got confused about the statement of the question.

What does the statement "$X,Y,Z$ were three disjoint subsets" mean? I understand it as $X \cap Y = \emptyset$, $X \cap Z = \emptyset$, $Y \cap Z = \emptyset$ and $X \cap Y \cap Z = \emptyset$.

Which makes my initial equations move towards things that don't make sense or are hard to move on. This is what I have so far:

$P(\mathcal{X}) = P(X \cup Y \cup Z)$ $P(X \cup Y \cup Z) = P(X) + P(Y) + P(Z) - P(X,Y) - P(Y,Z) - P(X,Z) + P(X,Y,Z)$

Using the info given:

$P(X \cup Y \cup Z) = P(X) + P(Y) + P(Z)$

Which gets me stuck. Thus, I am assuming that interpretation was wrong and instead I moved towards a different direction that didn't yield to much either...

$P(X \cup Y \cup Z) = P(X) + P(Y) + P(Z) - P(X,Y) - P(Y,Z) - P(X,Z) + P(X,Y,Z)$ $P(X \cup Y \cup Z) = P(X) + P(Y) + P(Z) - P(X|Y)P(Y) - P(Z)P(Y|Z) - P(Z)P(X|Z) + P(Z)P(X|Z)P(Y,Z)$

which then I try to factorize but I can't figure out how.

Some hints and clarification before stating the solution in your answer would be better (that way I can first have a clarification and try the proof again before reading the actual solution).

Thanks!

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This is self study, just for fun, for me :) No class no homework no nothing like that.

Answer 1

The puzzlement comes in great part from some horrendous abuses of notations.

The setting seems to be that one considers some disjoint sets $X$, $Y$ and $Z$ of random variables and the collection $\mathcal X$ which is the (disjoint) union of these three sets of random variables.

Then the hypothesis is that the random variables in $X$ are jointly independent of the random variables in $Y$ conditionally on the random variables in $Z$.

The conclusion one is asked to prove (in the discrete setting, to simplify the formulas) is that for every random variables $(X_i)$ in $X$, every random variables $(Y_j)$ in $Y$ and every random variables $(Z_k)$ in $Z$, the joint distribution of $((X_i),(Y_j),(Z_k))$ can be written as follows: $$P((X_i)=(x_i),(Y_j)=(y_j),(Z_k)=(z_k))=\phi_1((x_i),(z_k))\cdot\phi_2((y_j),(z_k)),$$ for some functions $\phi_1$ and $\phi_2$ and for every collections of (deterministic) values $(x_i)$, $(y_j)$, $(z_k)$.