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I came across this term in the deep learning book:

$p(x_{m+1}|x_1 ... x_m) = \int p(x_{m+1}|\theta)p(\theta|x_1 ... x_m)d\theta$

After some research I find that this term is the definition of the posterior predictive distribution. Is there a mathematical or intuitive proof of this? I'd prefer mathematical but anything helps right now.

EDIT: By proof, I am asking how LHS is equal to the RHS. For example, does this also work? If so, how do I arrive at PPD from the Bayes theorem?

$\frac{p(x_{m+1},x_1 ... x_m)}{p(x_1 ... x_m)} = \int p(x_{m+1}|\theta)p(\theta|x_1 ... x_m)d\theta$

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You can't arrive directly from Bayes Theorem because the model has further assumptions. That is, given the parameter set $\theta$, $x_{m+1}$ is independent of previous data; However, if not given we can infer from previous samples which is what PPD is trying to do actually. Without any assumptions, the integral should have been the following: $$p(x_{m+1}|x_1,...,x_m)=\int p(x_{m+1}|\theta,x_1,...,x_m)p(\theta|x_1,...,x_m)d\theta$$ But, we summarize $p(x_{m+1}|\theta,x_1,...,x_m)$ as $p(x_{m+1}|\theta)$ assuming conditional independence. Your link has also the following sentence:

Given the assumption that the observed and unobserved data are conditional independent given $\theta$.

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  • $\begingroup$ Thanks for the catch! I didn't notice that statement in the link. $\endgroup$ – SPRajagopal Sep 12 at 11:45
  • $\begingroup$ Followup question though: So, $x_{m+1}$ is not indenpendent of $x_1 ... x_m$; it's independent of $x_1 ... x_m$ given $\theta$. Is this correct? Because the former means that the LHS is just $p(x_{m+1})$. $\endgroup$ – SPRajagopal Sep 12 at 11:46
  • $\begingroup$ Yes, that's the whole idea. There is conditional independence, not full independence. $\endgroup$ – gunes Sep 12 at 11:52

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