# Is there a derivation for the Posterior Predictive Distribution?

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?

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

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 $$\theta$$.
• 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 $p(x_{m+1})$. Sep 12, 2019 at 11:46