0
$\begingroup$

I'm trying to find the posterior predictive distribution for data $X_i, \dots X_n$ from a a $Uniform [0, \theta]$ distribution. The prior distribution for $\theta$ is a $Pareto[\alpha, \beta]$ distribution:

I have found the posterior distribution to be: $\pi(\theta \vert \boldsymbol{X}) \propto Pareto[\alpha+n, \max(X_i, \beta)]$.

Now, when trying to find the posterior predictive distribution, I believe I have to find $$f(y|\boldsymbol{X}) = \int_{\Theta}{f(y|\theta)\pi(\theta|\boldsymbol{X})d\theta}$$

where y is the new data point.

I thought I might try the following:

$$f(y|\boldsymbol{X}) \propto \int_{\max(X_i, \beta)}^{\infty}\frac{1}{\theta}\frac{1}{\theta^{\alpha+n+1}}d\theta=\int_{\max(X_i, \beta)}^{\infty}\frac{1}{\theta^{\alpha+n+2}}d\theta$$

but I end up getting a weird answer, so I'm pretty sure this is incorrect. Could someone point me in the right direction? Thanks!

$\endgroup$
0
$\begingroup$

Your parameters were inverted, just in case you still care

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.