# Use of the Jeffreys prior in multidimensional models

Suppose a model,

$$x_{i} \sim N(\theta_{i}, \phi), \text{ for } i=1,\ldots,n$$

Furthermore, suppose the variance parameter, $\phi$, is some known constant.

The multidimensional Jeffreys prior is defined as,

$$\pi(\vec{\theta}) \propto \sqrt{\text{det}I(\vec{\theta})}$$

where $\text{det}I(\vec{\theta)}$ is the determinant of the multidimensional Fisher information.

Can a multidimensional Jeffreys prior be used to formulate a prior distribution for the posterior distribution, $p(\vec{\theta}|x_{1},\ldots,x_{n})$?

• Are the $x_i$ independent from each others? – peuhp Jun 11 '14 at 12:07
• @peuhp Yes, I meant to say that the $x_{i}$ are assumed to be i.i.d. – user9171 Jun 11 '14 at 12:31
• I am not sure to understand. How can they be i.i.d. if they are drawn from different distributions? – peuhp Jun 11 '14 at 12:39
• Sorry, you're absolutely right to be confused. That was a typo (bad habit, I'm afraid) - they're not identically distributed, just independent. – user9171 Jun 11 '14 at 12:43

If the $x_i$ are independent from each others, I would say that the Fisher information is diagonal and you will obtain a product of the univariate Jeffreys prior which in this case are the uniform ones. So your "multivariate prior" consists of : $$p_{\vec{\Theta}}(\vec{\theta}) \propto 1$$