Help with notation p(y|x, w, beta) In a Machine Learning course the notation used is the following:

I understand the normal (I think) on the RHS, but I can't figure out what this p is. Is it:


*

*Conditional distribution? If yes, why are there 3 parts after the |. I've only seen $f_{X|Y}(x|y)$ sort.

*Parametrized distribution of the $f(x; \theta)$ type? What is $\theta$?


Also, is it $p( (y|x),   w, \beta)$? Like is it y given x? or is it y given all of $x, w, \beta$?
This notation is prevalent in Bishop, PRML. Here is a section where the book describes this:

I've been learning stats on my own from All of Statistics (pdf), and I can't find equivalent notation (granted, I'm only on page 70).
Being a newbie to stats, I would appreciate any tips on this.
 A: It's conditional distribution. I find that in this area of math, authors are rather unforgiving in that there's a good handful of overlapping notations and conventions used, and you're expected to catch on.
Let's go back to the very basics and think of "events". $P(A|B)$ might help you understand something like $P(rain|cloudy)$, which would be higher than $P(rain|sunny)$. We can express $P(A|B) = P(A \cap B) / P(B)$.
Now you can generalize. $P(A|B,C)$ is the probability that $A$ happens given we know two things, specifically we know that $B$ and $C$ happen. In the case with events it's most like an "and" operator, so $P(A|B,C) = P(A|B \cap C)$, and it's only defined if $B$ and $C$ intersect.
Your case is more like $P(x|\theta)$. This breaks the intuition with "events" because $\theta$ isn't really an event. I find it tricky to use random variables like normal variables, and instead it's easy for me to temporarily right it as $p(x|\theta = \theta_0)$, where $\theta_0$ is a constant, not a random variable. Then it's clear to me that when we "condition" with the $|$ notation it's a lot like plugging in a value, or considering what happens if we plug in a value. Therefore you read this as: "What's the probability of $x$ once we plug in some value for $\theta$"? Plugging in $\theta$ is of course knowledge you'd like to use to figure out the probability of $x$.
Now in your case, $p(x|\theta_1, \theta_2)$ means, what's the probability of $x$ if we plug in, or "know", values for $\theta_1$ and $\theta_2$. Hopefully this way of describing things makes it clearer how to hop to the case of two conditioned variables or more.
