 Solution

The Jeffreys prior is 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,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.