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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?

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

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