Consider a Jeffreys prior where $p(\theta) \propto \sqrt{|i(\theta)|}$, where $i$ is the Fisher information.
I keep seeing this prior being mentioned as a uninformative prior, but I never saw an argument why it is uninformative. After all, it is not a constant prior, so there has to be some other argument.
I understand that it does not depends on reparametrization, which brings me to the next question. Is it that the determinant of the Fisher information does not depend on reparametrization? Because Fisher information definitely depends on the parametrization of the problem.
Thanks.
Best Answer
It's considered noninformative because of the parameterization invariance. You seem to have the impression that a uniform (constant) prior is noninformative. Sometimes it is, sometimes it isn't.
What happens with Jeffreys' prior under a transformation is that the Jacobian from the transformation gets sucked into the original Fisher information, which ends up giving you the Fisher information under the new parameterization. No magic (in the mechanics at least), just a little calculus and linear algebra.