I have a model involving two scalar parameters $\theta_1$ and $\theta_2$ and derived the Jeffreys prior for $\theta_1$ and $\theta_2$ independently (so for, e.g. $\pi(\theta_1)$, setting in the normalisation constant eveything that concern $\theta_2$, which in my case is possible). Then I computed the independent reference prior for $\theta_1$ and $\theta_2$ using the Theorem 2.1 (page 18) of this dissertation.
I found that these two are different.
- Is this possible ?
- If it is the case, is there any intuitive explanations for that ?