As title, what is the difference between $p(x,y\mid z)$ and $p(y\mid z,x)$, or they are the same? And more, is $p(x,y\mid z) p(z) = p(x,y,z)$ or $p(x,y\mid z) p(x,z) = p(x,y,z)$ ? Or should be $p(y\mid z,x)p(z,x) = p(x,y,z)$ ? Thanks for your answering.
Li
They are most definitely not the same thing. The notation $p(x|y)$ is read as "what is the probability of $x$ happening conditioned on $y$ having had happened?" So it is inherently clear that $p(x,y|z) \neq p(x|y,z)$. Read it out loud to yourself to see this is the case. Joint probability distributions are - in this sense - very nonassociative.