= Solution
For a regular one-parameter <sampling distribution>, the <Fisher information> and <Jeffreys prior> are
$$
I(\theta)=\mathbb E_\theta\left[\left(\frac{\partial}{\partial\theta}\log p_Y(Y\mid\theta)\right)^2\right],
\qquad \boxed{\pi_J(\theta)\propto\sqrt{I(\theta)}}.
$$
Under the usual differentiation and integrability conditions, $I(\theta)=-\mathbb E_\theta[\partial_\theta^2\log p_Y(Y\mid\theta)]$. This <prior distribution> transforms as a density under smooth one-to-one reparameterizations, so the rule is coordinate invariant. Its integral need not be finite; <posterior propriety> must still be established if it is an <improper prior>.
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