I have question about the following problem.
Suppose we have data $\left\{X_{i}, i=1, \ldots, n\right\}$ and assume the following hierarchical model. $$ \begin{array}{r} X_{i} \mid \Theta_{i}=\theta_{i} \stackrel{\text { ind }}{\sim} N\left(\theta_{i}, 1\right) \\ \Theta_{1}, \ldots, \Theta_{n} \mid T=\tau \stackrel{\mathrm{iid}}{\sim} N\left(0, \tau^{2}\right) \\ f_{T}(\tau) \propto \frac{1}{\tau} \end{array} $$
The prior on $\tau$ is improper and motivated as a Jeffreys' prior (note that the $N\left(0, \tau^{2}\right)$ belongs to a scale family). Investigate whether the posterior for $\tau$ is a proper density.
The following is what I have done:
I calculated the joint density of ${X_{i}, {\theta_{i}}}$, and $\tau$, which is: $$\frac{1}{\tau} \prod_{i=1}^{k}\left(\frac{1}{\sqrt{2 \pi}} e^{-\left(X_{i}-\theta_{i}\right)^{2} / 2} \frac{1}{\sqrt{2 \pi \tau^{2}}} e^{-\theta_{i}^{2} /\left(2 \tau^{2}\right)}\right)$$ I can see that theoretically we could integrate out all $\theta_{i}$ although it seems cumbersome since we have these exponentials with $\theta_{i}$ 's in them, but would we not get then the joint density of $\tau, X_{1}, \ldots, X_{k},$ whereas we want the conditional density of $\tau$ on these $X_{i}$ 's?