How do I parse P(A | B, C) and P(A, B | C)?

Question

Is there a universally accepted way of parsing the statement $P(A | B, C)$ where there is conditional and joint probability in the same expression? Should I read it as:

1. a conditional probability $P(A | (B,C))$ ? or
2. a joint probability $P( (A|B), C)$ ?

Likewise, how do I parse: $P(A, B|C)$ ?

Are there general rules? And if so, do they extend to 3+ variables?

Answer 1

Usually, $P(A|B,C)=P\left(A|(B,C)\right)$. And $P(A,B|C)=P\left((A,B)|C\right)$. So the general rule would be $P(A_1,...,A_n|B_1,...,B_m)=P\left((A_1,...,A_n)|(B_1,...,B_m)\right)$. Treat the things on the left and right hand sides of the vertical bar as their own groups with the commas meaning "and" or "intersection" ($\cap$). Unless you have a good reason to not parse the conditional probability that way, this is almost universally the way things are intended. I guess, the reason is because if you are given something, you're not going to be given it for some things and not other things. $P((A|B),C)$ implies that you have knowledge of B for A but not for C. But since they're part of the same sample space, you'd know that B happened for C too.

Answer 2

It is interpretation 1.

For 3+ variables $P(A_1 \mid A_2, A_3, \ldots, A_n)$ is read as "the probability of event $A_1$ occurring, given that events $A_2$ through $A_n$ are occurring."