Conditional and Joint distributions with many variables Suppose we have a probability function like:
$$
p(x|y,z)
$$
Does it mean it is a joint distribution with $x|y$ and $z$ or it is a conditional distribution of $x$ given $y,z$ ?
Intuitively, I think it is the second one, which is conditional distribution of $x$ on $y,z$ jointly.
What about the similar problem for the:
$$
p(x,y|z)
$$
?
 A: It is the second one. In the probability notation $p(\cdot|\cdot)$, normally whatever comes after the bar $|$ is assumed to be the events that occurred. In the case of $p(x|y,z)$, it can be interpreted as 1. the probability of $x$ given $y$ and $z$ or 2. the probability of $x$ when both $y$ and $z$ occurred.
A: If you expand the notation using the definition, we have:
$p(x | y, z) = \frac{p(x, y, z)}{p(y, z)} = \frac{p(X = x,\, Y = y, Z = z)}{p(Y = y,\, Z = z)}$, where I have made things a little more explicit in the last equality.
For simplicity of explanation, let's assume that each of the random variables is discrete. Then you can see that, as you intuitively thought, we are conditioning on the space of events where $Y = y$ and $Z = z$. You could also draw a Venn diagram of the situation, replacing the singleton sets $y$ and $z$ with some arbitrary events, and you could compare this to the case of a conditional distribution of the form $p(x | y)$, and you would see a visual confirmation of your intuition in the similarity of the diagrams.
