# Bayes' Theorem Application Question

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)?

• To understand why he used $A\mid p$ instead of $A$ you need to provide more context on what each variable is. Commented Jul 8, 2020 at 21:57
• I understand why he used A | p instead of A; what I don't understand is why P(data | A, x, y) isn't P(data | A, x, y, p) in the formula above.
– user722748
Commented Jul 9, 2020 at 0:13

## 1 Answer

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.