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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)?

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  • $\begingroup$ To understand why he used $A\mid p$ instead of $A$ you need to provide more context on what each variable is. $\endgroup$
    – Davi Barreira
    Commented Jul 8, 2020 at 21:57
  • $\begingroup$ 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. $\endgroup$
    – user722748
    Commented Jul 9, 2020 at 0:13

1 Answer 1

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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.

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