= Solution
{c}
The <Jeffreys prior> is \b[invariant under smooth one-to-one reparameterization]. This means that applying the rule directly in a new parameter $\eta=h(\theta)$ gives the same <prior distribution> as first constructing the rule for $\theta$ and then transforming its <prior density> by the change-of-variable formula.
More explicitly, for a differentiable invertible change of parameter with nonzero <derivative>,
$$
I_\eta(\eta)=I_\theta(\theta)\left(\frac{d\theta}{d\eta}\right)^2,
\qquad
\pi_{J,\eta}(\eta)\propto\sqrt{I_\eta(\eta)}
=\sqrt{I_\theta(\theta)}\left|\frac{d\theta}{d\eta}\right|.
$$
Thus $\pi_{J,\eta}(\eta)\,d\eta$ and $\pi_{J,\theta}(\theta)\,d\theta$ represent the same measure, up to the normalization constant. Invariance concerns the measure: the numerical values of the <prior densities> need not be equal in the two coordinates. The same change of variables then makes the <posterior distributions> agree. This is the <reparameterization invariance of Jeffreys prior>.
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