# Thread: Proof of semi-graphoid symmetry axiom

1. ## Proof of semi-graphoid symmetry axiom

I have some trouble understanding why this proof is a proof:
Lemma:
$I_p(X, Z, Y) \iff I_p(Y, Z, X)$

Proof:
$I_p(X, Z, Y) \iff P(C_x | C_y \land C_z) = P(C_x | C_z)$
Ok, this is just conditional independence, x independent of y given z.

$\iff \frac{P(C_x \land C_y \land C_z)}{P(C_y \land C_z)} = \frac{P(C_x \land C_z)}{P(C_z)}$
This is rewritten according to the rule of conditional independence P(A|B) = P(A & B) / P(B).

Bottom up the same is done for the right hand side of the lemma, but I don't get how the following and previous step are equal.. Why does this prove anything?
$\iff \frac{P(C_x \land C_y \land C_z)}{P(C_x \land C_z)} = \frac{P(C_y \land C_z)}{P(C_z)}$

$\iff P(C_y | C_x \land C_z) = P(C_y | C_z)$

$\iff I_p(Y, Z, X)$

2. ## Re: Proof of semi-graphoid symmetry axiom

Hey Lepzed.

Is the random variable X independent to Z?

3. ## Re: Proof of semi-graphoid symmetry axiom

The independence relation $I_p(X, Z, Y)$ expresses that in the context of information about Z, information about Y is irrelevant with respect to Y or put differently: X is condtionally independent of Y given Z.