Recently I asked a question and used the shorthand $P(A)$ for a random variable $A$ to quickly reason about conditional probabilities. However, I was informed that this must be written $P(A=a)$ for random variables and the notational shorthand does not make sense, and is only for events. Is this true?
In particular, I am interested in Bayesian Networks and want to write a factorisation such as $$P(A,B,C) = P(A)P(B\mid A)P(C\mid A,B) = P(A)P(B)P(C\mid A,B)$$ to say that the structure of the Bayesian Network is a graph with two root nodes corresponding to $A$ and $B$ which are not dependent on any other random variable and a third node corresponding to $C$ which is dependent on $A$ and $B$ (represented by directed edges).
Must I instead write
$$P(A=a,B=b,C=c) = P(A=a)P(B=b\mid A=a)P(C=c\mid A=a,B=b) = P(A=a)P(B=b)P(C=c\mid A=a,B=b)$$
to be mathematically coherent? Would I need to say this is for each $a,b,c$?
Edit: An example I am particularly interested in is on page 3 of this document, or equivalently the wikipedia entry for the chain rule of probability. Is it valid to write the chain rule of probability like this?