I have been trying to undertsand predictive posterior distribution.The following expression is given here
$p(x^*|x)=\int_\Theta c\times p(x^*,\theta|x)d\theta=\int_\Theta c\times p(x^*|\theta)p(\theta|x)d\theta$
I understand that the conditional probability is defined as
$f(A|B)=\frac{f(A,B)}{f(B)} = \frac{f(B|A)*f(A)}{f(B)}$
Can someone explain how the predictive posterior distribution is written using conditional probability?
You can compute the predictive posterior distribution as
$\begin{aligned} p(x^*|x) &= \int_{\Theta} p(x^*,\theta|x)d\theta \qquad ``\text{marginalizing out" }\theta\\ & = \int_{\Theta}p(x^*|\theta,x)p(\theta|x)d\theta \qquad \text{conditional probability rule: } P(A,B) = P(A|B) P(B)\\ &= \int_{\Theta}p(x^*|\theta)p(\theta|x)d\theta \qquad x^*\text{ is conditionally independent from }x \text{ given }\theta \end{aligned}$
where $p(\theta|x)$ is the posterior distribution of $\theta$.
The posterior predictive distribution will depend only on known quantities (the observed data $x$ and some prior's parameters).
