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I am having a brain freeze. Could you show the steps to get from line 1 to line 2?

Thanks!

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    $\begingroup$ You must have a condition: $D$ is independent of $\omega$ given $\theta$. $\endgroup$ – Zhanxiong Nov 15 '19 at 5:04
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In general, you can't. The second term in the first line, i.e. $p(\theta|\omega)$ corresponds to second and third terms in the second line, i.e. $p(\theta|\omega)p(\omega)$, which means you need to have $p(D|\theta,\omega)=P(D|\theta)$. It means $D,\omega$ are independent given $\theta$, as also noted in the comments. Probably, in your book, there is context indicating this information, e.g. a Bayes net or an experimental setup. A possible scenario is $D$ is your coin toss experiment, $\theta$ is your head probability of your coins. And $\omega$ is a parameter in the prior of $\theta$.

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  • $\begingroup$ Thanks. I guess I missed the part where a hierarchical model implicitly assumes D,ω are independent. Was trying my best to do algebra and couldn't get it work. $\endgroup$ – confused Nov 15 '19 at 14:39

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