I’m considering the following model: $$ H \sim \pi(H) \\ {\theta_n} | H \sim N\left(0_n, \left( \begin{array}{cccc} 1/H & 1 & \cdots & 1 \\ 1 & 1/H & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1/H \end{array}\right)\right) , $$ where $\theta_n$ is an $n$-dimensional random vector.
I’m assuming that $H\in(0,1)$. I’m wondering if there is a good prior $\pi$ for $H$.
I know that one can apply inverse gamma distribution for i.i.d. cases and inverse Wishart distribution for general multivariate cases. But in this case, I can’t find any good material which refers this issue.
Does anyone have a nice idea? I’m basically interesting in its posterior mean, so even if you can’t find the posterior distribution, that’s fine.
A beta distribution is a "typical" prior for parameter in $(0, 1)$. A special case is the uniform distribution on the range [0, 1].