# Interpretation of predictive posterior distribution

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?

• No, I am not able to understand how the expression for predictive posterior distribution is written Apr 20, 2018 at 17:13
• What exactly is that you don't understand? The quote gives what you seem to be asking for.
– Tim
Apr 20, 2018 at 18:40

\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).