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?