In certain cases, the Jeffreys prior for a full multidimensional model is generaly considered as inadequate, this is for example the case in: $$ y_i=\mu + \varepsilon_i \, , $$ (where $\varepsilon \sim N(0,\sigma^2)$, with $\mu$ and $\sigma$ unknown) where the following prior is prefered (to the full Jeffreys prior $\pi(\mu,\sigma)\propto \sigma^{-2}$): $$ p(\mu,\sigma) = \pi(\mu) \cdot \pi(\sigma) \propto \sigma^{-1}\, , $$ where $ \pi(\mu)$ is the Jeffreys prior obtained when keeping $\sigma$ fixed (and similarly for $p(\sigma)$). This prior coincides with the reference prior when treating $\sigma$ and $\mu$ in separate groups.
Question 1: Why does treating them as in separate groups make more sense than treating them in the same group (which will result, if I am correct (?), in the full dimensional Jeffreys prior, see [1]) ?
Then consider the following situation: $$ y_i=g(x_i,\mathbf{\theta}) +\varepsilon_i\, , $$ where $\theta \in \mathbb{R}^n$ is unknown, $\varepsilon_i \sim N(0,\sigma^2)$, $\sigma$ is unkown, and $g$ is a known non-linear function. In such a case, it is tempting and from my experience sometimes fruitful to consider the following decomposition: $$ p(\sigma,\theta)=\pi(\sigma) \pi(\theta) \, , $$ where $\pi(\sigma)$ and $\pi(\theta)$ are the Jeffreys prior for the two submodels as for the previous scale location example.
Question 2: In such a situation, can we say anything about the optimality (from an information theory perspective) of the derived prior $p(\sigma,\theta)$ ?
[1] From https://theses.lib.vt.edu/theses/available/etd-042299-095037/unrestricted/etd.pdf:
Finally, we note that Jeffreys' prior is a special case of a reference prior. Specifically, Jeffreys' prior corresponds to the reference prior in which all model parameters are treated in a single group.
