I'm having trouble verifying why the following is correct.
$p(x, y \mid z)= p(x \mid y, z) p(y \mid z)$
I tried grouping the $(x, y)$ together and split by the conditional, which gives me
$p(x, y \mid z) = p(z\mid x, y) p(x, y)/p(z)$
However, this did not bring me any closer. I'm uncertain about what kind of manipulations are allowed given more than 2 variables.
Say an expression like:
$p(a, b, c)$
Then I know from the chain rule that I can break it down to:
$p(a, b, c)=p(a \mid b, c) p(b, c) = p(a \mid b, c) p(b \mid c) p(c)$
Is it allowed to split by the second comma:
$p(a, b, c) = p(a, b \mid c) p(c) ?$
And even more complicated and expression like: $p(a|b,c)$
Am I allowed to rewrite this expression by grouping (a|b) together to give me something like $p(a|b,c)=p((a|b)|c)p(c)$ And does this expression even make sense?