I'm reading Networks by Mark Newman and I'm confused about one of his applications of Bayes' theorem. He begins with the following equation:
$$P(A, x, y \mid \text {data})=\frac{P(\text {data} \mid A, x, y) P(A) P(x) P(y)}{P(\text {data})}$$
He then assumes the prior probabilities $P(x)$ and $P(y)$ are uniform and therefore constants. He then states that we cannot assume the same for $P(A)$ and introduces the following as its prior probability $P(A|p)$(where $p$ is another probability on which $A$ depends)
From this he updates the formula above as follows:
$$P(A, x, y, p \mid \text {data})=\frac{P(\text {data} \mid A, x, y) P(A \mid p) P(p) P(x) P(y)}{P(\text {data})}$$
Why doesn't $P(\text{data}| A,x,y)$ include $p$ (and why did he introduce $p$ into the posterior probability)?
Note that $P(data)$ depends on $x,y,A$. So, $data$ is independent of $p$ when $A$ is given, hence, $P(data \mid A,p,x,y)= P(data \mid A,x,y)$.
For the second question, you need $p$ in the posterior to get the Bayes update with $A\mid p$, which you know the distribution.