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!

  • $\begingroup$ Your parameters were inverted, just in case you still care $\endgroup$ – Abraham Berriel Rangel Nov 28 '19 at 5:44

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